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(1)Heritable Edition. Geometric Aspects of General Topology Katsuro Sakai. ·.. ∞. 12 .ր. 0. ∆. January 31, 2013.

(2) Geometric Aspects of General Topology. c 2012 by Katsuro Sakai.

(3) Katsuro Sakai. Geometric Aspects of General Topology January 31, 2013. Springer.

(4) We can imagine and consider many mathematical concepts, such as numbers, spaces, maps, dimensions, etc., that can be indefinitely extended beyond infinity in our minds. Contemplating our mathematical ability in such a manner, I can recall this phrase from the Scriptures: Everything he has made pretty in its time. Even time indefinite he has put in their heart, that mankind may never find out the work that the true God has made from the start to the finish. — Ecclesiastes 3:11 May our Maker be glorified! Our brain is the work of his hands, as in Psalms 100:3, “Know that Jehovah is God. It is he that has made us, and not we ourselves.” There are many reasons to give thanks to God. Our mathematical ability is one of them. Note: Scripture quotations are from the modern-language New World Translation of the Holy Scriptures..

(5) Preface This book is designed for graduates studying Dimension Theory, ANR Theory (Theory of Retracts), and related topics. As is widely known, these two theories are connected with various fields in Geometric Topology as well as General Topology. So, for graduate students who wish to research subjects in General and Geometric Topology, understanding these theories will be valuable. Some excellent texts on these theories are the following: • • •. W. Hurewicz and H. Wallman, Dimension Theory, Princeton Univ. Press, Princeton, 1941; K. Borsuk, Theory of Retracts, MM 44, Polish Sci. Publ., Warsaw, 1966; S.-T. Hu, Theory of Retracts, Wayne State Univ. Press, Detroit, 1965.. However, these classical texts must be updated. This is the purpose of the present book. A comprehensive study of Dimension Theory may refer to the following book: •. R. Engelking, Theory of Dimensions, Finite and Infinite, SSPM 10, Heldermann Verlag, Lembo, 1995.. Engelking’s book, however, lacks results relevant to Geometric Topology. In this or any other textbook, no proof is given that dim X × I = dim X + 1 for a metrizable space X,1 and no example illustrates the difference between the small and large inductive dimensions or a hereditarily infinite-dimensional space (i.e., an infinite-dimensional space that has no finite-dimensional subspaces except for 0-dimensional subspaces).2 In the 1980s and 1990s, famous longstanding problems from Dimension Theory and ANR Theory were finally resolved. In the process, it became clear that these theories are linked with others. In Dimension Theory, the Alexandroff Problem had long remained unsolved. This problem queried the existence of an infinite-dimensional space whose cohomological dimension is finite. On the other hand, the CE Problem arose as a fascinating question in Shape Theory that asked whether there exists a cell-like map of a finitedimensional space onto an infinite-dimensional space. In the 1980s, it was 1. 2. This proof can be found in Kodama’s appendix of the following book: K. Nagami, Dimension Theory, Academic Press, Inc., New York, 1970. As will be mentioned later, a hereditarily infinite-dimensional space is treated in the book of J. van Mill: Infinite-Dimensional Topology.. i.

(6) shown that these two problems are equivalent. Finally, in 1988, by constructing an infinite-dimensional compact metrizable space whose cohomological dimension is finite, A.N. Dranishnikov solved the Alexandroff Problem. On the other hand, in ANR Theory, for many years it was unknown whether a metrizable topological linear space is an AR (or more generally, whether a locally equi-connected metrizable space is an ANR). In 1994, using a cell-like map of a finite-dimensional compact manifold onto an infinitedimensional space, R. Cauty constructed a separable metrizable topological linear space that is not an AR. These results are discussed in the latter half of the final chapter and provide an understanding of how deeply these theories are related to each other. This is also the purpose of this book. The notion of simplicial complexes is useful tool in Topology, and indispensable for studying both Theories of Dimension and Retracts. There are many textbooks from which we can gain some knowledge of them. Occasionally, we meet non-locally finite simplicial complexes. However, to the best of the author’s knowledge, no textbook discusses these in detail, and so we must refer to the original papers. For example, J.H.C. Whitehead’s theorem on small subdivisions is very important, but its proof cannot be found in any textbook. This book therefore properly treats non-locally finite simplicial complexes. The homotopy type of simplicial complexes is usually discussed in textbooks on Algebraic Topology using CW complexes, but we adopt a geometrical argument using simplicial complexes, which is easily understandable. As prerequisites for studying infinite-dimensional manifolds, Jan van Mill provides three chapters on simplicial complexes, dimensions, and ANRs in the following book: •. J. van Mill, Infinite-Dimensional Topology, Prerequisites and Introduction, North-Holland Math. Library 43, Elsevier Sci. Publ. B.V., Amsterdam, 1989.. These chapters are similar to the present book in content, but they are introductory courses and restricted to separable metrizable spaces. The important results mentioned above are not treated except for an example of a hereditarily infinite-dimensional space. Moreover, one can find an explanation of the Alexandroff Problem and the CE Problem in Chapter 3 of the following book: •. A. Chigogidze, Inverse Spectra, North-Holland Math. Library 53, Elsevier Sci. B.V., Amsterdam, 1996.. Unfortunately, this book is, however, inaccessible for graduate students. The present text has been in use by the author for his graduate class at the University of Tsukuba. Every year, a lecture has been given based on some topic selected from this book except the final chapter, and the same material has been used for an undergraduate seminar. Readers are required to finish the initial courses of Set Theory and General Topology. Basic knowledge of Linear Algebra is also a prerequisite. Except for the latter half of the final chapter, this book is self-contained. ii.

(7) Chapter 1 develops the general material relating to topological spaces appropriate for graduate students. It provides a supplementary course for students who finished an undergraduate course in Topology. We discuss paracompact spaces and some metrization theorems for non-separable spaces that are not treated in a typical undergraduate course.3 This chapter also contains Michael’s theorem on local properties, which can be applied in many situations. We further discuss the direct limits of towers (increasing sequences) of spaces, which are appear in Geometric and Algebraic Topology.4 A nonHausdorff direct limit of a closed tower of Hausdorff spaces is included. The author has not found any literature representing such an example. The limitation topology on the function spaces is also discussed. Chapter 2 is devoted to topological linear spaces and convex sets. There are many good textbooks on these subjects. This chapter represents a short course on fundamental results on them. First, we establish the existing relations between these objects and to General and Geometric Topology. Convex sets are then discussed in detail. This chapter also contains Michael’s selection theorem. Moreover, we show the existence of free topological spaces. In Chapter 3, simplicial complexes are treated without assuming local finiteness. As mentioned above, we provide proof of J.H.C. Whitehead’s theorem on small subdivisions. The simplicial mapping cylinder is introduced and applied to prove the Whitehead–Milnor theorem on the homotopy type of simplicial complexes. It is also applied to prove that every weak homotopy equivalence between simplicial complexes is a homotopy equivalence. The inverse limits of inverse sequences are also discussed, and it is shown that every completely metrizable space is homeomorphic to locally finite-dimensional simplicial complexes with the metric topology. These results cannot be found in any other book dealing with simplicial complexes but are buried in old journals. D.W. Henderson established the metric topology version of the Whitehead theorem on small subdivisions, but his proof is valid only for locally finitedimensional simplicial complexes. Here we offer a complete proof without the assumption of local finite-dimensionality. Knowledge of homotopy groups is not required, even when weak homotopy equivalences are discussed. However, we do review homotopy groups in Appendix 3.14 because they are helpful in the second half of Chapter 6. Chapters 4 and 5 are devoted to Dimension Theory and ANR Theory, respectively. We prove basic results and fundamental theorems on these theo3. 4. These subjects are discussed in Munkres’ book, now a very popular textbook at the senior or the first-year graduate level: J.R. Munkres, Topology, 2nd ed., Prentice Hall, Inc., Upper Saddle River, 2000. The direct limits are discussed in Appendix of Dugundji’s book: J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, 1966. But, they are not discussed even in Engelking’s book, a comprehensive reference book for General Topology: R. Engelking, General Topology, Revised and completed edition, SSPM 6, Heldermann Verlag, Berlin, 1989.. iii.

(8) ries. The contents are very similar to Chapters 4 and 5 of van Mill’s “InfiniteDimensional Topology”. However, as mentioned previously, we do not restrict ourselves to separable metrizable spaces and instead go on to prove further results. In Chapter 4, we describe a non-separable metrizable space such that the large inductive dimension does not coincide with the small inductive dimension. As mentioned above, such an example is not treated in any other textbook on Dimension Theory (not even Engelking’s book). Here, we present Kulesza’s example with Levin’s proof. The transfinite inductive dimension is also discussed, which is not treated in van Mill’s book. Further, we prove that every completely metrizable space with dimension n is homeomorphic to the inverse limit of an inverse sequence of metric simplicial complexes with dimension n. Finally, hereditarily infinite-dimensional spaces are discussed based on van Mill’s book. In Chapter 5, we discuss topics that are not treated in van Mill’s book or in the two classical books by Hu and Borsuk mentioned above. Following are examples of such topics: uniform ANRs in the sense of Michael and its completion; Kozlowski’s theorem that the metrizable range of a fine homotopy equivalence of an ANR is also an ANR; Cauty’s characterization, with Sakai’s proof, that a metrizable space is an ANR if and only if every open set has the homotopy type of an ANR; Haver’s theorem that every countable-dimensional locally contractible metrizable space is an ANR; and Bothe’s theorem, with Kodama’s proof, that every n-dimensional metrizable space can be embedded in an (n + 1)-dimensional AR as a closed set. In Chapter 6, cell-like maps and related topics are discussed. The first half is self-contained, but the second half is not because some algebraic results are necessary. In the first half, we examine the existing relations between cell-like maps, soft maps, fine homotopy equivalences, etc. The second half is devoted to related topics. In particular, the CE Problem is explained and Cauty’s example is presented. Note that Chigogidze’s “Inverse Spectra” is the only book dealing with soft maps and provides an explanation of the Alexandroff Problem and the CE Problem. In the second half of Chapter 6, using the K-theory result of Adams, we present the Taylor example. Eilenberg–MacLane spaces are usually constructed as CW complexes, but here they are constructed as simplicial complexes. To avoid using cohomology, we define the cohomological dimension geometrically. By applying the cohomological dimension, we can prove the equality dim X × I = dim X + 1 for every metrizable space X. We also discuss the Alexandroff Problem and the CE Problem as mentioned above. The equivalence of these problems is proved. Next, we describe the Dydak–Walsh example that gives an affirmative answer to the Alexandroff Problem. However, this part of the text is not self-contained. As a corollary, we answer the CE Problem, i.e., we obtain a cell-like mapping of a finite-dimensional compact manifold onto an infinite-dimensional compactum. We also present Cauty’s example, i.e., a metrizable topological linear space that is not an abiv.

(9) solute extensor. In the proof, we need the above cell-like mapping to be open, and we therefore use Walsh’s open mapping approximation theorem. A proof of Walsh’s theorem is beyond the scope of this book. The author would like to express his sincere appreciation to his teacher, Professor Yukihiro Kodama, who introduced him to Shape Theory and Infinite-Dimensional Topology and warmly encouraged him to persevere in his study. He owes his gratitude to Ross Geoghegan for improving the written English text. He is also grateful to Haruto Ohta, Taras Banakh and Zhongqiang Yang for their valuable comments and suggestions. Finally, he also warmly thanks his graduate students, Yutaka Iwamoto, Yuji Akaike, Shigenori Uehara, Masayuki Kurihara, Masato Yaguchi, Kotaro Mine, Atsushi Yamashita, Minoru Nakamura, Atsushi Kogasaka, Katsuhisa Koshino, and Hanbiao Yang for their careful reading and helpful comments.. Katsuro Sakai December 2012 Tsukuba. This work was supported by JSPS KAKENHI Grant Number 22540063.. v.

(10) Geometric Aspects of General Topology. Katsuro Sakai. vi.

(11) Contents. 0. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.1 Terminology and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 Banach Spaces in the Product of Real Lines . . . . . . . . . . . . . . . . 11. 1. Metrization and Paracompact Spaces . . . . . . . . . . . . . . . . . . . . . . 1.1 Products of Compact Spaces and Perfect Maps . . . . . . . . . . . . . . 1.2 The Tietze Extension Theorem and Normalities . . . . . . . . . . . . . 1.3 Stone’s Theorem and Metrization . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Sequences of Open Covers and Metrization . . . . . . . . . . . . . . . . . 1.5 Complete Metrizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Paracompactness and Local Properties . . . . . . . . . . . . . . . . . . . . . 1.7 Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 The Direct Limits of Towers of Spaces . . . . . . . . . . . . . . . . . . . . . 1.9 The Limitation Topology for Spaces of Maps . . . . . . . . . . . . . . . . 1.10 Counter-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2. Topology of Linear Spaces and Convex Sets . . . . . . . . . . . . . . . . 73 2.1 Flats and Affine Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.2 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.3 The Hahn–Banach Extension Theorem . . . . . . . . . . . . . . . . . . . . . 87 2.4 Topological Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.5 Finite-Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.6 Metrizability and Normability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2.7 The Closed Graph and Open Mapping Theorems . . . . . . . . . . . . 120 2.8 Continuous Selections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 2.9 Free Topological Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129. 3. Simplicial Complexes and Polyhedra . . . . . . . . . . . . . . . . . . . . . . . 135 3.1 Simplexes and Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.2 Complexes and Subdivisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 3.3 Product Complexes and Homotopy Extension . . . . . . . . . . . . . . . 152. 21 21 26 31 35 40 45 52 56 62 66.

(12) 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14. PL Maps and Simplicial Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 The Metric Topology of Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . 164 Derived and Barycentric Subdivisions . . . . . . . . . . . . . . . . . . . . . . 174 Small Subdivisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Admissible Subdivisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 The Nerves of Open Covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 The Inverse Limits of Metric Polyhedra . . . . . . . . . . . . . . . . . . . . 206 The Mapping Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 The Homotopy Type of Simplicial Complexes . . . . . . . . . . . . . . . 223 Weak Homotopy Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Appendix: Homotopy Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233. 4. Dimensions of Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 4.1 The Brouwer Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . 251 4.2 Characterizations of Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 4.3 Dimension of Metrizable Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 4.4 Fundamental Theorems on Dimension . . . . . . . . . . . . . . . . . . . . . . 271 4.5 Inductive Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 4.6 Infinite Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 4.7 Compactification Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 4.8 Embedding Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 4.9 Universal Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 4.10 N¨ obeling Spaces and Menger Compacta . . . . . . . . . . . . . . . . . . . . 303 4.11 Total Disconnectedness and the Cantor Set . . . . . . . . . . . . . . . . . 310 4.12 Totally Disconnected Spaces with dim = 0 . . . . . . . . . . . . . . . . . . 315 4.13 Examples of Infinite-Dimensional Spaces . . . . . . . . . . . . . . . . . . . 321 4.14 Appendix: The Hahn–Mazurkiewicz Theorem . . . . . . . . . . . . . . . 323. 5. Retracts and Extensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 5.1 The Dugundji Extension Theorem and ANEs . . . . . . . . . . . . . . . 337 5.2 Embeddings of Metric Spaces and ANRs . . . . . . . . . . . . . . . . . . . 346 5.3 Small Homotopies and LEC Spaces . . . . . . . . . . . . . . . . . . . . . . . . 353 5.4 The Homotopy Extension Property . . . . . . . . . . . . . . . . . . . . . . . . 359 5.5 Complementary Pairs of ANRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 5.6 Realizations of Simplicial Complexes . . . . . . . . . . . . . . . . . . . . . . . 370 5.7 Fine Homotopy Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 5.8 Completions of ANRs and Uniform ANRs . . . . . . . . . . . . . . . . . . 383 5.9 Homotopy Types of Open Sets in ANRs . . . . . . . . . . . . . . . . . . . . 390 5.10 Countable-Dimensional ANRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 5.11 The Local n-Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 5.12 Finite-Dimensional ANRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 5.13 Embeddings into Finite-Dimensional ARs . . . . . . . . . . . . . . . . . . . 415. viii.

(13) 6. Cell-like Maps and Related Topics . . . . . . . . . . . . . . . . . . . . . . . . . 425 6.1 Trivial Shape and Related Properties . . . . . . . . . . . . . . . . . . . . . . 425 6.2 Soft Maps and the 0-Dimensional Selection Theorem . . . . . . . . . 431 6.3 Hereditary n-Equivalence and Local Connections . . . . . . . . . . . . 437 6.4 Fine Homotopy Equivalences between ANRs . . . . . . . . . . . . . . . . 444 6.5 Hereditary Shape Equivalences and U V n Maps . . . . . . . . . . . . . . 448 6.6 The Near-Selection Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 6.7 The Suspensions and the Taylor Example . . . . . . . . . . . . . . . . . . . 456 6.8 The Simplicial Eilenberg–MacLane Complexes . . . . . . . . . . . . . . 466 6.9 Cohomological Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 6.10 Alexandroff’s Problem and the CE Problem . . . . . . . . . . . . . . . . 485 6.11 Free Topological Linear Spaces over Compacta . . . . . . . . . . . . . . 494 6.12 A Non-AR Metric Linear Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 500. Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517. ix.

(14) Geometric Aspects of General Topology. Katsuro Sakai. x.

(15) 0 Preliminaries. The reader should have finished a first course in Set Theory and General Topology; basic knowledge of Linear Algebra is also a prerequisite. First, we introduce some terminology and notation, before explaining the concept of Banach spaces contained in the product of real lines.. 0.1 Terminology and Notation For the standard sets, we use the following notation: • • • • • • • •. N — the set of natural numbers (i.e., positive integers); ω = N ∪ {0} — the set of non-negative integers; Z — the set of integers; Q — the set of rationals; R = (−∞, ∞) — the real line with the usual topology; C — the complex plane; R+ = [0, ∞); I = [0, 1] — the unit closed interval.. A (topological) space is assumed to be Hausdorff and a map is a continuous function. A singleton is a space consisting of one point, which is also said to be degenerate. A space is said to be non-degenerate if it is not a singleton. Let X be a space and A ⊂ X. We denote • • • •. clX A (or cl A) — the closure of A in X; intX A (or int A) — the interior of A in X; bdX A (or bd A) — the boundary of A in X; idX (or id) — the identity map of X.. For spaces X and Y , •. X ≈ Y means that X and Y are homeomorphic..

(16) 2. 0 Preliminaries. Given subspaces X1 , . . . , Xn ⊂ X and Y1 , . . . , Yn ⊂ Y , • •. (X, X1 , . . . , Xn ) ≈ (Y, Y1 , . . . , Yn ) means that there exists a homeomorphism h : X → Y such that h(X1 ) = Y1 , . . . , h(Xn ) = Yn ; (X, x0 ) ≈ (Y, y0 ) means (X, {x0 }) ≈ (Y, {y0 }).. We call (X, x0 ) a pointed space and x0 its base point. For a set Γ , the cardinality of Γ is denoted by card Γ . The weight w(X), the density dens X, and the cellurality c(X) of a space X are defined as follows: • • •. w(X) = min{card B | B is an open basis for X}; dens X = min{card D | D is a dense set in X}; c(X) = sup{card G | G is a pair-wise disjoint open collection}.. As is easily observed, c(X) dens X w(X) in general. If X is metrizable, all these cardinalities coincide. Indeed, let D be a dense set in X with card D = dens X, and G be a pairwise disjoint collection of non-empty open sets in X. Since each G ∈ G meets D, we have an injection g : G → D, hence card G card D = dens X. dens X. Now, let B be an open basis for X with It follows that c(X) card B = w(X). By taking any point xB ∈ B from each B ∈ B, we have a dense set {xB | B ∈ B} in X, which implies dens X w(X). When X is metrizable, we show the converse inequality. The case card X < ℵ0 is trivial. We may assume that X = (X, d) is a metric 1 and card X ℵ0 . Let D be a dense set in space with diam X X with card D = dens X. Then, {B(x, 1/n) | x ∈ D, n ∈ } is an open basis for X, which implies w(X) dens X. For each n ∈ , using Zorn’s Lemma, we can find a maximal 2−n -discrete subset Xn ⊂ X, 2−n for every pair of distinct points x, y ∈ Xn . Then, i.e., d(x, y) Gn = {B(x, 2−n−1 ) | x ∈ Xn } is a pairwise disjoint open collection, and c(X). Observe that X∗ = n∈Xn is hence we have card Xn = card Gn dense in X, which implies supn∈card Xn = card X∗ dens X. Therefore, c(X) dens X.. . . . . . Ë. For the product space γ∈Γ Xγ , the γ-coordinate of each point x ∈ is denoted by x(γ), i.e., x = (x(γ))γ∈Γ . For each γ ∈ Γ , the projecγ∈Γ Xγ tion prγ : γ∈Γ Xγ → X γ is defined by prγ (x) = x(γ). For Λ ⊂ Γ , the projection prΛ : γ∈Γ Xγ → λ∈Λ Xλ is defined by prΛ (x) = x|Λ (= (x(λ))λ∈Λ ). In the case that Xγ = X for every γ ∈ Γ , we write γ∈Γ Xγ = X Γ . In particular, X N is the product space of countable infinite copies of X. When Γ = {1, . . . , n}, X Γ = X n is the product space of n copies of X. For the product space X × Y , we denote the projections by prX : X × Y → X and prY : X × Y → Y . A compact metrizable space is called a compactum and a connected compactum is called a continuum.1 For a metrizable space X, we denote 1. Their plurals are compacta and continua, respectively..

(17) 0.1 Terminology and Notation. •. 3. Metr(X) — the set of all admissible metrics of X.. Now, let X = (X, d) be a metric space, x ∈ X, ε > 0, and A, B ⊂ X. We use the following notation: • Bd (x, ε) = y ∈ X d(x, y) < ε — the ε-neighborhood of x in X (or the open ball with center x and radius ε); • Bd (x, ε) = y ∈ X d(x, y) ε — the closed ε-neighborhood of x in X (or the closed ball with center x and radius ε); • Nd (A, ε) = x∈A B (x, ε) d — the ε-neighborhood of A in X; d(x, y) • diamd A = sup x, y ∈ A — the diameter of A; of x from A; • d(x, A) = inf d(x, y) y ∈ A — the distance • distd (A, B) = inf d(x, y) x ∈ A, y ∈ B — the distance of A and B.. It should be noted that Nd ({x}, ε) = Bd (x, ε) and d(x, A) = distd ({x}, A). For a collection A of subsets of X, let • meshd A = sup diamd A A ∈ A — the mesh of A.. If there is no possibility of confusion, we can drop the subscript d and write B(x, ε), B(x, ε), N(A, ε), diam A, dist(A, B), and mesh A. The standard spaces are listed below: •. • • • • • • • •. Rn — the n-dimensional Euclidean space with the norm . x = x(1)2 + · · · + x(n)2 ,. 0 = (0, . . . , 0) ∈ Rn — the origin, the zero vector or the zero element, ei ∈ Rn — the unit vector defined by ei (i) = 1 and ei (j) = 0 for j = i; n . x. = 1 Sn−1 = x ∈ R — the unit (n − 1)-sphere; n n B = x ∈ R x 1 — the unit closed n-ball; n+1 ∆n = x ∈ (R+ )n+1 i=1 x(i) = 1 — the standard n-simplex; Q = [−1, 1]N — the Hilbert cube; — the space of sequences; s = RN ∞ i μ0 = xi ∈ {0, 1} — the Cantor (ternary) set; i=1 2xi /3 ν 0 = R \ Q — the space of irrationals; 2 = {0, 1} — the discrete space of two points.. Note that Sn−1 , Bn , and ∆n are not product spaces, even though the same notations are used for product spaces. The indexes n − 1 and n represent their dimensions (the indexes of μ0 and ν 0 are identical). As is well-known, the countable product 2N of the discrete space 2 = {0, 1} is homeomorphic to the Cantor set μ0 by the correspondence: x →. 2x(i) i∈N. 3i. .. On the other hand, the countable product NN of the discrete space N of natural numbers is homeomorphic to the space ν 0 of irrationals. In fact, NN ≈.

(18) 4. 0 Preliminaries. (0, 1) \ Q ≈ (−1, 1) \ Q ≈ ν 0 . These three homeomorphisms are given as follows: x →. 1. ;. 1. x(1) +. t → 2t − 1;. 1. x(2) +. x(3) +. s →. s . 1 − |s|. 1 ... .. That the first correspondence is a homeomorphism can be verified as follows: For each n ∈ , let an : → I be a map defined by 1. an (x) =. .. 1. x(1) + x(2) +. 1 ... . +. 1 x(n). Then, 0 < a2 (x) < a4 (x) < · · · < a3 (x) < a1 (x) 1. Using the fact shown below, we can conclude that the first correspondence ∋ x → α(x) = limn→∞ an (x) ∈ (0, 1) is well-defined and continuous. Fact For every m > n, |an (x) − am (x)| < (n + 1)−1 . This fact can be shown by induction on n ∈ . First, observe that |a1 (x) − a2 (x)| =. 1 < 12 , x(1)(x(1)x(2) + 1). which implies the case n = 1. When n > 1, for each x ∈ , define x∗ ∈ by x∗ (i) = x(i + 1). By the inductive assumption, |an−1 (x∗ ) − am−1 (x∗ )| < n−1 for m > n, which gives us |an−1 (x∗ ) − am−1 (x∗ )| (x(1) + an−1 (x∗ ))(x(1) + am−1 (x∗ )) |an−1 (x∗ ) − am−1 (x∗ )| (1 + an−1 (x∗ ))(1 + am−1 (x∗ )) |an−1 (x∗ ) − am−1 (x∗ )| < 1 + |an−1 (x∗ ) − am−1 (x∗ )|. |an (x) − am (x)| =. 1 n−1 = . 1 + n−1 n+1 Let t = q1 /q0 ∈ (0, 1) ∩ , where q1 < q0 ∈ . Since q0 /q1 = t−1 > 1, we can choose x1 ∈ so that x1 q0 /q1 < x1 +1. Then, 1/(x1 +1) < t 1/x1 . If t = 1/x1 , then x1 < q0 /q1 , and hence t−1 = q0 /q1 = x1 + q2 /q1 for some q2 ∈ with q2 < q1 . Now, we choose x2 ∈ so that x2 q1 /q2 < x2 + 1. x1 + 1/x2 , so 1/(x1 + 1/x2 ) t< Thus, x1 + 1/(x2 + 1) < x1 + q2 /q1 1/(x1 + 1/(x2 + 1)). If t = 1/(x1 + 1/x2 ), then x2 < q1 /q2 . Similarly, we.

(19) 0.1 Terminology and Notation. 5. write q1 /q2 = x2 + q3 /q2 , where q3 ∈ with q3 < q2 (< q1 ), and choose q2 /q3 < x3 + 1. Then, 1/(x1 + 1/(x2 + 1/x3 )) t< x3 ∈ so that x3 1/(x1 + 1/(x2 + 1/(x3 + 1))). This process has only a finite umber of steps (at most q1 steps). Thus, we have the following unique representation: 1. t=. ,. 1. x1 + x2 +. x1 , . . . , x n ∈ .. 1 ... . +. 1 xn. It follows that α() ⊂ (0, 1) \ . For each t ∈ (0, 1) \ , choose x1 ∈ so that x1 < t−1 < x1 + 1. Then, 1/(x1 + 1) < t < 1/x1 and t−1 = x1 + t1 for some t1 ∈ (0, 1) \ . Next, choose x2 ∈ so that x2 < t−1 1 < x2 + 1. Thus, x1 + 1/(x2 + 1) < x1 + t1 < x1 + 1/x2 , and so 1/(x1 + 1/x2 ) < t < 1/(x1 + 1/(x2 + 1)). Again, write −1 t−1 1 = x2 + t2 , t2 ∈ (0, 1) \ , and choose x3 ∈ so that x3 < t2 < x3 + 1. Then, 1/(x1 + 1/(x2 + 1/(x3 + 1))) < t < 1/(x1 + 1/(x2 + 1/x3 )). We can iterate this process infinitely many times. Thus, there is the unique x = (xn )n∈ ∈ such that a2n (x) < t < a2n+1 (x) for each n ∈ , where α(x) = limn→∞ an (x) = t. Therefore, α : → (0, 1) \ is a bijection. In the above, let a2n (x) < s < a2n−1 (x) and define y = (yi )i∈ ∈ for this s similar to x for t. Then, α(y) = s and xi = yi for each i 2n − 1, i.e., the first 2n − 1 coordinates of x and y are all the same. This means that α−1 is continuous.. Let f : A → Y be a map from a closed set A in a space X to another space Y . The adjunction space Y ∪f X is the quotient space (X ⊕ Y )/∼, where X ⊕ Y is the topological sum and ∼ is the equivalence relation corresponding to the decomposition of X ⊕ Y into singletons {x}, x ∈ X \ A, and sets {y} ∪ f −1(y), y ∈ Y (the latter is a singleton {y} if y ∈ Y \ f (A)). In the case that Y is a singleton, Y ∪f X ≈ X/A. One should note that, in general, the adjunction spaces are not Hausdorff. Some further conditions are necessary for the adjunction space to be Hausdorff. Let A and B be collections of subsets of X and Y ⊂ X. We define • • •. A ∧ B = {A ∩ B | A ∈ A, B ∈ B}; A|Y = {A ∩ Y | A ∈ A}; A[Y ] = {A ∈ A | A ∩ Y = ∅}.. When each A ∈ A is contained in some B ∈ B, it is said that that A refines B and denoted by: A ≺ B or B ≻ A. It is said that A covers Y (or A is a cover of Y in X) if Y ⊂ A (= A∈A A). When Y = X, a cover of Y in X is simply called a cover of X. A cover of Y in X is said to be open or closed in X depending on whether its members are open or closed in X. If A is an open cover of X then A|Y is an open cover of.

(20) 6. 0 Preliminaries. Y and A[Y ] is an open cover of Y in X. When A and B are open covers of X, A ∧ B is also an open cover of X. For covers A and B of X, it is said that A is a refinement of B if A ≺ B, where A is an open (or closed) refinement if A is an open (or closed) cover. For a space X, we denote •. cov(X) — the collection of all open covers of X.. Let (Xγ )γ∈Γ be a family of (topological) spaces and X = γ∈Γ Xγ . The weak topology on X with respect to (Xγ )γ∈Γ is defined as follows: U ⊂ X is open in X ⇔ ∀γ ∈ Γ, U ∩ Xγ is open in Xγ. A ⊂ X is closed in X ⇔ ∀γ ∈ Γ, A ∩ Xγ is closed in Xγ .. Suppose that X has the weak topology with respect to (Xγ )γ∈Γ , and that the topologies of Xγ and Xγ ′ agree on Xγ ∩ Xγ ′ for any γ, γ ′ ∈ Γ . If Xγ ∩ Xγ ′ is closed (resp. open) in Xγ for any γ, γ ′ ∈ Γ then each Xγ is closed (resp. open) in X and the original topology of each Xγ is a subspace topology inherited from X. In the case that Xγ ∩

(21) Xγ ′ = ∅ for γ = γ ′ , X is the topological sum of (Xγ )γ∈Γ , denoted by X = γ∈Γ Xγ . Let f : X → Y be a map. For A ⊂ X and B ⊂ Y , we denote f (A) = f (x) x ∈ A and f −1 (B) = x ∈ X f (x) ∈ B .. For collections A and B of subsets of X and Y , respectively, we denote f (A) = f (A) A ∈ A and f −1 (B) = f −1 (B) B ∈ B .. The restriction of f to A ⊂ X is denoted by f |A. It is said that a map g : A → Y extends over X if there is a map f : X → Y such that f |A = g. Such a map f is called an extension of g. Let [a, b] be a closed interval, where a < b. A map f : [a, b] → X is called a path (from f (a) to f (b)) in X, and we say that two points f (a) and f (b) are connected by the path f in X. An embedding f : [a, b] → X is called an arc (from f (a) to f (b)) in X, and the image f ([a, b]) is also called an arc. Namely, a space is called an arc if it is homeomorphic to I. It is known that each pair of distinct points x, y ∈ X are connected by an arc if and only if they are connected by a path.2 For spaces X and Y , we denote •. C(X, Y ) — the set of (continuous) maps from X to Y .. For maps f, g : X → Y (i.e., f, g ∈ C(X, Y )), • 2. f ≃ g means that f and g are homotopic (or f is homotopic to g), This will be shown in Corollary 4.14.6..

(22) 0.1 Terminology and Notation. 7. that is, there is a map h : X × I → Y such that h0 = f and h1 = g, where ht : X → Y , t ∈ I, are defined by ht (x) = h(x, t), and h is called a homotopy from f to g (between f and g). When g is a constant map, it is said that f is null-homotopic, which we denote by f ≃ 0. The relation ≃ is an equivalence relation on C(X, Y ). The equivalence class [f ] = {g ∈ C(X, Y ) | g ≃ f } is called the homotopy class of f . We denote •. [X, Y ] = {[f ] | f ∈ C(X, Y )} = C(X, Y )/ ≃ — the set of the homotopy classes of maps from X to Y .. For each f, f ′ ∈ C(X, Y ) and g, g ′ ∈ C(Y, Z), we have the following: f ≃ f ′ , g ≃ g ′ ⇒ gf ≃ g ′ f ′ . Thus, we have the composition [X, Y ] × [Y, Z] → [X, Z] defined by ([f ], [g]) → [g][f ] = [gf ]. Moreover, •. X ≃ Y means that X and Y are homotopy equivalent (or X is homotopy equivalent to Y ),3. that is, there are maps f : X → Y and g : Y → X such that gf ≃ idX and f g ≃ idY , where f is called a homotopy equivalence and g is a homotopy inverse of f . Given subspaces X1 , . . . , Xn ⊂ X and Y1 , . . . , Yn ⊂ Y , a map f : X → Y is said to be a map from (X, X1 , . . . , Xn ) to (Y, Y1 , . . . , Yn ), written f : (X, X1 , . . . , Xn ) → (Y, Y1 , . . . , Yn ), if f (X1 ) ⊂ Y1 , . . . , f (Xn ) ⊂ Yn . We denote. •. C((X, X1 , . . . , Xn ), (Y, Y1 , . . . , Yn )) — the set of maps from (X, X1 , . . . , Xn ) to (Y, Y1 , . . . , Yn ).. A homotopy h between maps f, g ∈ C((X, X1 , . . . , Xn ), (Y, Y1 , . . . , Yn )) requires the condition that ht ∈ C((X, X1 , . . . , Xn ), (Y, Y1 , . . . , Yn )) for every t ∈ I, i.e., h is regarded as the map h : (X × I, X1 × I, . . . , Xn × I) → (Y, Y1 , . . . , Yn ). Thus, ≃ is an equivalence relation on C((X, X1 , . . . , Xn ), (Y, Y1 , . . . , Yn )). We denote •. [(X, X1 , . . . , Xn ), (Y, Y1 , . . . , Yn )] = C((X, X1 , . . . , Xn ), (Y, Y1 , . . . , Yn ))/≃.. When there exist maps f : (X, X1 , . . . , Xn ) → (Y, Y1 , . . . , Yn ), g : (Y, Y1 , . . . , Yn ) → (X, X1 , . . . , Xn ). such that gf ≃ idX and f g ≃ idY , we denote 3. It is also said that X and Y have the same homotopy type or X has the homotopy type of Y ..

(23) 8. •. 0 Preliminaries. (X, X1 , . . . , Xn ) ≃ (Y, Y1 , . . . , Yn ).. Similarly, for each pair of pointed spaces (X, x0 ) and (Y, y0 ), • • •. C((X, x0 ), (Y, y0 )) = C((X, {x0 }), (Y, {y0 })); [(X, x0 ), (Y, y0 )] = C((X, x0 ), (Y, y0 ))/ ≃; (X, x0 ) ≃ (Y, y0 ) means (X, {x0 }) ≃ (Y, {y0 }).. For A ⊂ X, a homotopy h : X × I → Y is called a homotopy relative to A if h({x} × I) is degenerate (i.e., a singleton) for every x ∈ A. When a homotopy from f to g is a homotopy relative to A (where f |A = g|A), we denote •. f ≃ g rel. A.. Let f, g : X → Y be maps and U a collection of subsets of Y (in usual, U ∈ cov(Y )). It is said that f and g are U-close (or f is U-close to g) if {f (x), g(x)} x ∈ X ≺ U ∪ {y} y ∈ Y ,. which implies that U covers the set {f (x), g(x) | f (x) = g(x)}. A homotopy h is called a U-homotopy if h({x} × I) x ∈ X ≺ U ∪ {y} y ∈ Y ,. which implies that U covers the set h({x} × I) h({x} × I) is non-degenerate .. We say that f and g are U-homotopic (or f is U-homotopic to g) and denoted by f ≃U g if there is a U-homotopy h : X × I → Y such that h0 = f and h1 = g. When Y = (Y, d) is a metric space, we define the distance between f, g ∈ C(X, Y ) as follows: d(f, g) = sup d(f (x), g(x)) x ∈ X .. In general, it may be possible that d(f, g) = ∞, in which case d is not a metric on the set C(X, Y ). If Y is bounded or X is compact, then this d is a metric on the set C(X, Y ), called the sup-metric. For ε > 0, we say that f and g are ε-close or f is ε-close to g if d(f, g) < ε. A homotopy h is called an ε-homotopy if mesh{h({x} × I) | x ∈ X} < ε, where f = h0 and g = h1 are said to be ε-homotopic and denoted by f ≃ε g. In the above, even if d is not a metric on C(X, Y ) (i.e., d(f, g) = ∞ for some f, g ∈ C(X, Y )), it induces a topology on C(X, Y ) such that each f has a neighborhood basis consisting of Bd (f, ε) = g ∈ C(X, Y ) d(f, g) < ε , ε > 0. which is called the uniform convergence topology..

(24) 0.1 Terminology and Notation. 9. The compact-open topology on C(X, Y ) is generated by the sets K; U = f ∈ C(X, Y ) f (K) ⊂ U ,. where K is any compact set in X and U is any open set in Y . With respect to this topology, we have the following: 0.1.1 Proposition Every map f : Z × X → Y (or f : X × Z → Y ) induces the map f¯ : Z → C(X, Y ) defined by f¯(z)(x) = f (z, x) (or f¯(z)(x) = f (x, z)). Proof. For each z ∈ Z, it is easy to see that f¯(z) : X → Y is continuous, i.e., f¯(z) ∈ C(X, Y ). Thus, f¯ is well-defined. To verify the continuity of f¯ : Z → C(X, Y ), it suffices to show that −1 ¯ f (K; U ) is open in Z for each compact set K in X and each open set U in Y . Let z ∈ f¯−1 (K; U ), i.e., f ({z} × K) ⊂ U . Using the compactness of K, we can easily find an open neighborhood V of z in Z such that f (V × K) ⊂ U , ⊔ which means that V ⊂ f¯−1 (K; U ). ⊓ With regards to the relation ≃ on C(X, Y ), we have the following: 0.1.2 Proposition For each f, g ∈ C(X, Y ), if f ≃ g then f and g are connected by a path in C(X, Y ). If X is metrizable or locally compact, the converse is also true, that is, f ≃ g if and only if f and g are connected by a path in C(X, Y ).4 Proof. By Proposition 0.1.1, a homotopy h : X × I → Y from f to g induces ¯ : I → C(X, Y ) defined as ¯h(t)(x) = h(x, t) for each t ∈ I and the path h ¯ ¯ x ∈ X, where h(0) = f and h(1) = g. For a path ϕ : I → C(X, Y ) from f to g, we define the homotopy ϕ˜ : X × I → Y as ϕ(x, ˜ t) = ϕ(t)(x) for each (x, t) ∈ X × I. Then, ϕ˜0 = ϕ(0) = f and ϕ˜1 = ϕ(1) = g. It remains to show that ϕ˜ is continuous if X is metrizable or locally compact. In the case that X is locally compact, for each (x, t) ∈ X × I and for each open neighborhood U of ϕ(x, ˜ t) = ϕ(t)(x) in Y , x has a compact neighborhood K in X such that ϕ(t)(K) ⊂ U , i.e., ϕ(t) ∈ K; U . By the continuity of ϕ, t has a neighborhood V in I such that ϕ(V ) ⊂ K; U . Thus, K × V is a neighborhood of (x, t) ∈ X × I and ϕ(K ˜ × V ) ⊂ U . Hence, ϕ˜ is continuous. In the case that X is metrizable, let us assume that ϕ˜ is not continuous at (x, t) ∈ X × I. Then, ϕ(x, ˜ t) has some open neighborhood U in Y such that ϕ(V ˜ ) ⊂ U for any neighborhood V of (x, t) in X × I. Let d ∈ Metr(X). For each n ∈ N, there exist xn ∈ X and tn ∈ I such that d(xn , x) < 1/n, |tn − t| < 1/n and ϕ(x ˜ n , tn ) ∈ U . Because xn → x (n → ∞) and ϕ(t) is continuous, we have n0 ∈ N such that ϕ(t)(xn ) ∈ U for all n n0 . Note 4. More generally, this is valid for every k-space X, where X is a -space provided U is open in X if U ∩ K is open in K for every compact set K ⊂ X. A k-space is also called a compactly generated space..

(25) 10. 0 Preliminaries. that K = {xn , x | n n0 } is compact and ϕ(t)(K) ⊂ U . Because tn → t (n → ∞) and ϕ is continuous at t, ϕ(tn1 )(K) ⊂ U for some n1 n0 . Thus, ⊔ ϕ(x ˜ n1 , tn1 ) ∈ U , which is a contradiction. Consequently, ϕ˜ is continuous. ⊓ Remark 1. It is easily observed that Proposition 0.1.2 is also valid for C((X, X1 , . . . , Xn ), (Y, Y1 , . . . , Yn )). 0.1.3 Some Properties of the Compact-Open Topology The following hold with respect to the compact-open topology: (1) For f ∈ C(Z, X) and g ∈ C(Y, Z), the following are continuous: f ∗ : C(X, Y ) → C(Z, Y ), f ∗ (h) = h ◦ f ; g∗ : C(X, Y ) → C(X, Z), g∗ (h) = g ◦ h.. (2) In the case that Y is locally compact, the following (composition) is continuous: C(X, Y ) × C(Y, Z) ∋ (f, g) → g ◦ f ∈ C(X, Z). Sketch of Proof. Let K be a compact set in X and U an open set in Z with f ∈ C(X, Y ) and g ∈ C(Y, Z) such that g ◦ f (K) ⊂ U . Since Y is locally compact, we have an open set V in Y such that cl V is compact, f (K) ⊂ V and g(cl V ) ⊂ U . Then, f ′ (K) ⊂ V and g ′ (cl V ) ⊂ U imply g ′ ◦ f ′ (K) ⊂ U .. (3) For each x0 ∈ X, the following (evaluation) is continuous: C(X, Y ) ∋ f → f (x0 ) ∈ Y. (4) When X is locally compact, the following (evaluation) is continuous: C(X, Y ) × X ∋ (f, x) → f (x) ∈ Y. In this case, for every map f : Z → C(X, Y ), the following is continuous: Z × X ∋ (z, x) → f (z)(x) ∈ Y. (5) In the case that X is locally compact, w(Y ) w(C(X, Y )) ℵ0 w(X)w(Y ). Sketch of Proof. By embedding Y into C(X, Y ), we obtain the first inequality. For the second, we take open bases BX and BY for X and Y , respectively, such that card BX = w(X), card BY = w(Y ), and cl A is compact for every A ∈ BX . The following is an open sub-basis for C(X, Y ): ¬ ¨ © B =

(26) cl A, B ¬ (A, B) ∈ BX × BY . Indeed, let K be a compact set in X, U be an open set in Y , and f ∈ C(X, Y ) with f (K) ⊂ U , i.e., f ∈

(27) K, U . First, find B1 , . . . , Bn ∈ BY so that f (K) ⊂ B1 ∪ · · · ∪ Bn ⊂ U . Next, find A1 , . . . , Am ∈ BX so that K ⊂ AÌ1 ∪ · · · ∪ Am and each cl Ai is contained in some f −1 (Bj(i) ). Then, f ∈ m i=1

(28) cl Ai , Bj(i) ⊂

(29) K, U ..

(30) 0.2 Banach Spaces in the Product of Real Lines. 11. (6) If X is compact and Y = (Y, d) is a metric space, then the sup-metric on C(X, Y ) is admissible for the compact-open topology on C(X, Y ). Sketch of Proof. Let K be a compact set in X and U be an open set in Y with f ∈ C(X, Y ) such that f (K) ⊂ U . Then, δ = dist(f (K), Y \ U ) > 0, and d(f, f ′ ) < δ implies f ′ (K) ⊂ U . Conversely, for each ε > 0 and f ∈ C(X, Y ), we have x1 , . . . , xn ∈ X such that X = n −1 (B(f (xi ), ε/4). Observe that i=1 f f ′ (f −1 (B(f (xi ), ε/4))) ⊂ B(f(xi ), ε/2) (∀i = 1, . . . , n) ⇒ d(f, f ′ ) < ε.. (7) Let X = n∈N Xn , where Xn is compact and Xn ⊂ int Xn+1 . If Y = (Y, d) is a metric space, then C(X, Y ) with the compact-open topology is metrizable. Sketch of Proof. We define a metric ρ on C(X, Y ) as follows:. . . ρ(f, g) = sup min n−1 , sup d(f (x), g(x)) . n∈. . x∈Xn. Then, ρ is admissible for the compact-open topology on C(X, Y ). To see this, refer to the proof of (6).. 0.2 Banach Spaces in the Product of Real Lines Throughout this section, let Γ be an infinite set. We denote •. Fin(Γ ) — the set of all non-empty finite subsets of Γ .. Note that card Fin(Γ ) = card Γ . The product space RΓ is a linear space with the following scalar multiplication and addition: RΓ × R ∋ (x, t) → tx = (tx(γ))γ∈Γ ∈ RΓ ;. RΓ × RΓ ∋ (x, y) → x + y = (x(γ) + y(γ))γ∈Γ ∈ RΓ . In this section, we consider various (complete) norms defined on linear subspaces of RΓ . In general, the unit closed ball and the unit sphere of a normed linear space X = (X, · ) are denoted by BX and SX , respectively. Namely, let BX = x ∈ X x 1 and SX = x ∈ X x = 1 . The zero vector (the zero element) of X is denoted by 0X , or simply 0 if there is no possibility of confusion. Before considering norms, we first discuss the product topology of RΓ . The scalar multiplication and addition are continuous with respect to the product topology. Namely, RΓ with the product topology is a topological linear space.5 Note that w(RΓ ) = card Γ . 5. Refer to §2.4..

(31) 12. 0 Preliminaries Let B0 be a countable open basis for open basis:. ¨Ì. γ∈F. Ê.. Then,. ¬¬. ÊΓ. has the following. ©. pr−1 γ (Bγ ) F ∈ Fin(Γ ), Bγ ∈ B0 (γ ∈ F ) .. Thus, we have w(ÊΓ ) ℵ0 card Fin(Γ ) = card Γ . Let B be an open basis for ÊΓ . For each B ∈ B, we can find FB ∈ Fin(Γ ) such that prγ (B) = Ê for every γ ∈ Γ \ FB . Then, card B∈B FB ℵ0 card B. If card B < card Γ then card B∈B FB < card Γ , so we have γ0 ∈ Γ \ B∈B FB . The open set Γ pr−1 contains some B ∈ B. Then, prγ0 (B) ⊂ (0, ∞), which γ0 ((0, ∞)) ⊂ Ê card Γ , means that γ0 ∈ FB . This is a contradiction. Therefore, card B and thus we have w(ÊΓ ) card Γ .. Ë. Ë. Ë. . . For each γ ∈ Γ , we define the unit vector eγ ∈ RΓ by eγ (γ) = 1 and eγ (γ ′ ) = 0 for γ ′ = γ. It should be noted that {eγ | γ ∈ Γ } is not a Hamel basis for RΓ , and the linear span of {eγ | γ ∈ Γ } is the following:6 RΓf = x ∈ RΓ x(γ) = 0 except for finitely many γ ∈ Γ ,. N which is a dense linear subspace of RΓ . The subspace RN f of s = R is also denoted by sf , which is the space of finite sequences (with the product topology). When card Γ = ℵ0 , the space RΓ is linearly homeomorphic to the space of sequences s = RN , i.e., there exists a linear homeomorphism between RΓ and s, where the linear subspace RΓf is linearly homeomorphic to sf by the same homeomorphism. The following fact can easily be observed:. Fact The following are equivalent: (a) (b) (c) (d). RΓ is metrizable; RΓf is metrizable; RΓf is first countable; card Γ ℵ0 .. The implication (c) ⇒ (d) is shown as follows: Let {Ui | i ∈ Æ} be a neighborhood basis of 0 in ÊΓf . Then, each Γi = {γ ∈ Γ | Êeγ ⊂ Ui } is finite. If Γ is uncountable then Γ \ i∈Γi = ∅, i.e., Êeγ ⊂ i∈Ui for some γ ∈ Γ . In this case, Ui ⊂ pr−1 γ ((−1, 1)) for every i ∈ Æ, which is a contradiction.. Ë. Ì. Thus, every linear subspace L of RΓ containing RΓf is non-metrizable if Γ is uncountable, and it is metrizable if Γ is countable. On the other hand, due to the following proposition, every linear subspaces L of RΓ containing RΓf is non-normable if Γ is infinite. 0.2.1 Proposition Let Γ be an infinite set. Any norm on RΓf does not induce the topology inherited from the product topology of RΓ . 6. The linear subspace generated by a set B is called the linear span of B..

(32) 0.2 Banach Spaces in the Product of Real Lines. 13. Proof. Assume that the topology of RΓf is induced by a norm · . Because U = {x ∈ RΓf | x < 1} is an open neighborhood of 0 in RΓf , we have a finite set F ⊂ Γ and neighborhoods Vγ of 0 ∈ R, γ ∈ F , such that RΓf ∩ pr−1 (V ) ⊂ U . Take γ0 ∈ Γ \ F . As Reγ0 ⊂ U , we have eγ0 −1 eγ0 ∈ U γ∈F γ −1γ but eγ0 eγ0 = eγ0 −1 eγ0 = 1, which is a contradiction. ⊓ ⊔. The Banach space ℓ∞ (Γ ) and its closed linear subspaces c(Γ ) ⊃ c0 (Γ ) are defined as follows: • ℓ∞ (Γ ) = x ∈ RΓ supγ∈Γ |x(γ)| < ∞ with the sup-norm. x ∞ = sup |x(γ)|; γ∈Γ. • •. c(Γ ) = x ∈ RΓ ∃t ∈ R such that ∀ε > 0, |x(γ) − t| < ε except for finitely many γ ∈ Γ ; c0 (Γ ) = x ∈ RΓ ∀ε > 0, |x(γ)| < ε except for finitely many γ ∈ Γ .. These are linear subspaces of RΓ , but are not topological subspace according to Proposition 0.2.1. The space c(Γ ) is linearly homeomorphic to c0 (Γ ) × R by the correspondence c0 (Γ ) × R ∋ (x, t) → (x(γ) + t)γ∈Γ ∈ c(Γ ). This correspondence and its inverse are Lipschitz with respect to the norm (x, t) = max{x∞ , |t|}. Indeed, let y = (x(γ) + t)γ∈Γ . Then, y∞ x∞ + |t| 2(x, t). Because |t| |y(γ)| + |x(γ)| y∞ + |x(γ)| y∞ . Moreover, for every γ ∈ Γ and x ∈ 0 (Γ ), it follows that |t| |x(γ)| |y(γ)| + |t| 2y∞ for every γ ∈ Γ . Hence, x∞ 2y∞ , and thus we have (x, t) 2y∞ .. Furthermore, we denote RΓf with this norm as ℓf∞ (Γ ). We then have the inclusions: ℓf∞ (Γ ) ⊂ c0 (Γ ) ⊂ c(Γ ) ⊂ ℓ∞ (Γ ).. The topology of ℓf∞ (Γ ) is different from the topology inherited from the product topology. Indeed, {eγ | γ ∈ Γ } is discrete in ℓf∞ (Γ ), but 0 is a cluster point of this set with respect to the product topology. We must pay attention to the following fact: 0.2.2 Proposition For an arbitrary infinite set Γ , w(ℓ∞ (Γ )) = 2card Γ but w(c(Γ )) = w(c0 (Γ )) = w(ℓf∞ (Γ )) = card Γ. Proof. The characteristic map χΛ : Γ → {0, 1} ⊂ R of Λ ⊂ Γ belongs to ℓ∞ (Γ ) (χ∅ = 0 ∈ ℓ∞ (Γ )), where χΛ − χΛ′ ∞ = 1 if Λ = Λ′ ⊂ Γ . It follows that w(ℓ∞ (Γ )) = c(ℓ∞ (Γ )) 2card Γ . Moreover, since QΓ ∩ ℓ∞ (Γ ) is dense in ℓ∞ (Γ ), we have.

(33) 14. 0 Preliminaries Γ w(ℓ∞ (Γ )) = dens ℓ∞ (Γ ) card QΓ = ℵcard = 2card Γ . 0. On the other hand, eγ ∈ ℓf∞ (Γ ) for each γ ∈ Γ and eγ − eγ ′ ∞ = 1 if γ = γ ′ . Since ℓf∞ (Γ ) ⊂ c0 (Γ ), it follows that w(c0 (Γ )) w(ℓf∞ (Γ )) = c(ℓf∞ (Γ )) card Γ. Moreover, c0 (Γ ) has the following dense subset: QΓf = x ∈ QΓ x(γ) = 0 except for finitely many γ ∈ Γ ,. and so it follows that. w(c0 (Γ )) = dens c0 (Γ ) card QΓf ℵ0 card Fin(Γ ) = card Γ. Thus, we have w(c0 (Γ )) = w(ℓf∞ (Γ )) = card Γ . As already observed, c(Γ ) ≈ c0 (Γ ) × R, hence w(c(Γ )) = w(c0 (Γ )). ⊓ ⊔ When Γ = N, we write • • • •. ℓ∞ (N) = ℓ∞ — the space of bounded sequences, c(N) = c — the space of convergent sequences, c0 (N) = c0 — the space of sequences convergent to 0, and ℓf∞ (N) = ℓf∞ , — the space of finite sequences with the sup-norm,. where ℓf∞ = sf as (topological) spaces. According to Proposition 0.2.2, c and c0 are separable, but ℓ∞ is non-separable. When card Γ = ℵ0 , the spaces ℓ∞ (Γ ), c(Γ ), and c0 (Γ ) are linearly isometric to these spaces ℓ∞ , c and c0 , respectively. a directed set by ⊂. For x ∈ RΓ , we say that Here, we regard Fin(Γ ) as γ∈F x(γ) F ∈Fin(Γ ) is convergent, and define γ∈Γ x(γ) is convergent if. x(γ) =. x(γ) =. γ∈Γ. lim. F ∈Fin(Γ ). x(γ).. x(γ).. γ∈F. In the case that x(γ) 0 for all γ ∈ Γ , γ∈Γ x(γ) is convergent if and only if γ∈F x(γ) F ∈Fin(Γ ) is upper bounded, and then γ∈Γ. sup. F ∈Fin(Γ ) γ∈F. By this reason, γ∈Γ x(γ) < ∞ means that γ∈Γ x(γ) is convergent. For x ∈ RN , we should distinguish i∈N x(i) from ∞ x(i). When the n ∞ i=1 x(i) is convergent, is convergent, we say that x(i) sequence i=1 i=1 n∈N and define n ∞. x(i). x(i) = lim i=1. n→∞. i=1.

(34) 0.2 Banach Spaces in the Product of Real Lines. 15. ∞ Evidently, if i∈N x(i) is convergent, then i=1 x(i) is also convergent and ∞ i∈N x(i) is not necessary convergent even i∈N x(i). However, i=1 x(i) = ∞ if i=1 x(i) is convergent. In fact, due to Proposition 0.2.3 below, we have the following:. i∈N. x(i) is convergent ⇔. ∞. i=1. |x(i)| is convergent.. 0.2.3 PropositionFor an infinite set Γ and x ∈ RΓ , γ∈Γ x(γ) is conver|x(γ)| < ∞. In this case, Γx = {γ ∈ Γ | x(γ) = 0} is gent if and only if ∞ γ∈Γ countable, and γ∈Γ x(γ) = i=1 x(γi ) for any sequence (γi )i∈N in Γ such that Γx ⊂ {γi | i ∈ N} and γi = γj if i = j. Proof. Let us denote Γ+ = {γ ∈ Γ | x(γ) > 0} and Γ− = {γ ∈ Γ | x(γ) < 0}. Then, Γx = Γ+ ∪ Γ− . If γ∈Γ x(γ) is convergent, we have F0 ∈ Fin(Γ ) such that . x(γ) − x(γ) < 1. F0 ⊂ F ∈ Fin(Γ ) ⇒ γ∈Γ. γ∈F. Then, for each E ∈ Fin(Γ+ ) ∪ Fin(Γ− ), . |x(γ)| = x(γ) = x(γ) − x(γ) < 2. γ∈E\F0. γ∈E∪F0. γ∈E\F0. γ∈F0. Hence, γ∈F |x(γ)| < γ∈F0 |x(γ)| + 4 for every F ∈ Fin(Γ ), which means that γ∈Γ |x(γ)| < ∞. γ∈F |x(γ)| F ∈Fin(Γ ) is upper bounded, i.e., |x(γ)| < ∞. Then, Conversely, we assume that γ∈Γ for each n ∈ N, Γn = {γ ∈ Γ | |x(γ)| > 1/n} is finite, and hence Γ = x n∈N Γn is countable. Note that γ∈Γ+ |x(γ)| < ∞ and γ∈Γ− |x(γ)| < ∞. We show that. x(γ) = |x(γ)| − |x(γ)|. γ∈Γ. γ∈Γ+. γ∈Γ−. For each ε > 0, we can find F+ ∈ Fin(Γ+ ) and F− ∈ Fin(Γ− ) such that. |x(γ)|. |x(γ)| |x(γ)| − ε/2 < F± ⊂ E ∈ Fin(Γ± ) ⇒ γ∈Γ±. γ∈E. γ∈Γ±. Then, it follows that, for each F ∈ Fin(Γ ) with F ⊃ F+ ∪ F− , . x(γ) − |x(γ)| − |x(γ)| γ∈F γ∈Γ+ γ∈Γ− . |x(γ)| |x(γ)| − |x(γ)| + |x(γ)| − γ∈F ∩Γ+. < ε/2 + ε/2 = ε.. γ∈Γ+. γ∈F ∩Γ−. γ∈Γ−.

(35) 16. 0 Preliminaries. Now, let (γi )i∈N be a sequence in Γ such that Γx ⊂ {γi | i ∈ N} and γi = γj if i = j. We define n0 = max{i ∈ N | γi ∈ F+ ∪ F− }. For each n n0 , it follows from F+ ∪ F− ⊂ {γ1 , . . . , γn } that n . x(γi ) − |x(γ)| < ε. |x(γ)| − i=1. Thus, we also have. •. γ∈Γ−. γ∈Γ+. . γ∈Γ. x(γ) =. ∞. i=1. ⊓ ⊔. x(γi ).. For each p 1, the Banach space ℓp (Γ ) is defined as follows: ℓp (Γ ) = x ∈ RΓ γ∈Γ |x(γ)|p < ∞ with the norm. x p =. . γ∈Γ. p. |x(γ)|. 1/p. .. Similar to ℓf∞ (Γ ), we denote the space RΓf with this norm by ℓfp (Γ ). The triangle inequality for the norm xp is known as the Minkowski inequality, which is derived from the following H¨ older inequality:. . 1/p. a γ bγ. apγ. γ∈Γ. γ∈Γ. 1 1−1/p. bγ. 1−1/p. for every aγ , bγ. 0.. γ∈Γ.

(36) .

(37)

(38) . Indeed, for every x, y ∈ ℓp (Γ ), x + ypp =. |x(γ) + y(γ)|p. γ∈Γ. |x(γ)| + |y(γ)| |x(γ) + y(γ)|p−1. γ∈Γ. =. |x(γ)| · |x(γ) + y(γ)|p−1 +. γ∈Γ. |y(γ)| · |x(γ) + y(γ)|p−1. γ∈Γ. 1/p. |x(γ)|p. γ∈Γ. 1−1/p. 1 (p−1) 1−1/p. |x(γ) + y(γ)|. γ∈Γ. 1/p. p. |y(γ)|. +. γ∈Γ. = xp + yp so it follows that x + yp. 1 (p−1) 1−1/p. |x(γ) + y(γ)|. γ∈Γ. x +. 1−1/p ypp. xp + yp .. = xp + yp.

(39). 1−1/p. x + ypp , x + yp.

(40) 0.2 Banach Spaces in the Product of Real Lines. 17. As for c0 (Γ ), we can show w(ℓp (Γ )) = card Γ . When card Γ = ℵ0 , the Banach space ℓp (Γ ) is linearly isometric to ℓp = ℓp (N), which is separable. The space ℓ2 (Γ ) is the Hilbert space with the inner product. x, y = x(γ)y(γ), γ∈Γ. which is well-defined because. |x(γ)y(γ)| 21 ( x 22 + y 22 ) < ∞. γ∈Γ. For 1 p < q, we have ℓp (Γ ) ℓq (Γ ) c0 (Γ ) as sets (or linear spaces). These inclusions are continuous because x ∞ x q x p for every x ∈ ℓp (Γ ). When Γ is infinite, the topology of ℓp (Γ ) is distinct from that induced by the norm · q or · ∞ (i.e., the topology inherited from ℓq (Γ ) or c0 (Γ )). In fact, the unit sphere Sℓp (Γ ) is closed in ℓp (Γ ) but not closed in ℓq (Γ ) for any q > p, nor in c0 (Γ ). To see this, take distinct γi ∈ Γ , i ∈ N, and let (xn )n∈N be the sequence in Sℓp (Γ ) defined by xn (γi ) = n−1/p for i n and xn (γ) = 0 for γ = γ1 , . . . , γn . It follows that xn ∞ = n−1/p → 0 (n → ∞) and 1/q. xn q = n · n−q/p = n(p−q)/pq → 0 (n → ∞) because (p − q)/pq < 0. For 1 p ∞, we have RΓf ⊂ ℓp (Γ ) as sets (or linear spaces). We denote by ℓfp (Γ ) this RΓf with the topology inherited from ℓp (Γ ), and we write ℓfp (N) = ℓfp (when Γ = N). From Proposition 0.2.1, we know ℓfp (Γ ) = RΓf as spaces for any infinite set Γ . In the above, the sequence (xn )n∈N is contained in the unit sphere Sℓfp (Γ ) of ℓfp (Γ ), which means that Sℓfp (Γ ) is not closed in ℓfq , hence ℓfp = ℓfq as spaces for 1 p < q ∞. Note that Sℓfp (Γ ) is a closed subset of ℓfq for 1 q < p. Concerning the convergence of sequences in ℓp (Γ ), we have the following: 0.2.4 Proposition For each p ∈ N and x ∈ ℓp (Γ ), a sequence (xn )n∈N converges to x in ℓp (Γ ) if and only if. x p = lim xn p and x(γ) = lim xn (γ) for every γ ∈ Γ . n→∞. n→∞. Proof. The “only if” part is trivial, so we concern ourselves with proving the “if” part for ℓp (Γ ). For each ε > 0, we have γ1 , . . . , γk ∈ Γ such that. γ =γi. |x(γ)|p = x pp −. k. i=1. |x(γi )|p < 2−p εp /4.. Choose n0 ∈ N so that if n n0 then xn pp − x pp < 2−p εp /8,.

(41) 18. 0 Preliminaries. |xn (γi )|p − |x(γi )|p < 2−p εp /8k and |xn (γi ) − x(γi )|p < εp /4k. for each i = 1, . . . , k. Then, it follows that. γ =γi. |xn (γ)|p = xn pp −. xn pp. =. k. i=1. |xn (γi )|p. − x pp + x pp −. k. i=1. |x(γi )|p. k. + |x(γi )|p − |xn (γi )|p i=1. < 2−p εp /8 + 2−p εp /4 + 2−p εp /8 = 2−p εp /2, and hence we have. xn − x pp . k. i=1. |xn (γi ) − x(γi )|p +. p. < ε /4 + p. γ =γi p. p. p. γ =γi. 2 |xn (γ)| + p. p. < ε /4 + ε /2 + ε /4 = ε , that is, xn − x p < ε.. p 2p max |xn (γ)|, |x(γ)|. γ =γi. 2p |x(γ)|p. ⊓ ⊔. Remark 2. It should be noted that Proposition 0.2.4 is valid not only for sequences, but also for nets, which means that the unit spheres Sℓp (Γ ) , p ∈ N, are subspaces of the product space RΓ , whereas RΓ and RΓf are not metrizable if Γ is uncountable. Therefore, if 1 p < q ∞, then Sℓp (Γ ) is also a subspace of ℓq (Γ ), although, as we have seen, Sℓp (Γ ) of ℓp (Γ ) is not closed in the space ℓq (Γ ). The unit sphere Sℓfp (Γ ) of ℓfp (Γ ) is a subspace of RΓf (⊂ RΓ ), and also a subspace of ℓq (Γ ) for 1 q ∞. Remark 3. The “if” part of Proposition 0.2.4 does not hold for the space c0 (Γ ) (although the “only if” part obviously does hold), where Γ is infinite. For instance, take distinct γn ∈ Γ , n ∈ ω, and let (xn )n∈N be the sequence in c0 (Γ ) defined by xn = eγn + eγ0 . Then, xn ∞ = 1 for each n ∈ N, lim xn (γ0 ) = 1 = eγ0 (γ0 ) and. n→∞. lim xn (γ) = 0 = eγ0 (γ) for γ = γ0 ,. n→∞. but xn − eγ0 ∞ = 1 for every n ∈ Γ . In addition, the unit sphere Sc0 (Γ ) of c0 (Γ ) is not a subspace of RΓ , because eγn ∈ Sc0 (Γ ) but (eγn )n∈N converges to 0 in RΓ . Concerning the topological classification of ℓp (Γ ), we have the following: 0.2.5 Theorem (Mazur) For each 1 < p < ∞, ℓp (Γ ) is homeomorphic to ℓ1 (Γ ). By the same homeomorphism, ℓfp (Γ ) is also homeomorphic to ℓf1 (Γ )..

(42) 0.2 Banach Spaces in the Product of Real Lines. 19. Proof. We define ϕ : ℓ1 (Γ ) → ℓp (Γ ) and ψ : ℓp (Γ ) → ℓ1 (Γ ) as follows: ϕ(x)(γ) = sign x(γ) · |x(γ)|1/p for x ∈ ℓ1 (Γ ), ψ(x)(γ) = sign x(γ) · |x(γ)|p for x ∈ ℓp (Γ ),. where sign 0 = 0 and sign a = a/|a| for a = 0. We can apply Proposition 0.2.4 to verify the continuity of ϕ and ψ. In fact, the following functions are continuous: 1/p ∈ R, ℓ1 (Γ ) ∋ x → ϕ(x)(γ) ∈ R, γ ∈ Γ ; ℓ1 (Γ ) ∋ x → ϕ(x) p = x 1 p ℓp (Γ ) ∋ x → ψ(x) 1 = x p ∈ R, ℓp (Γ ) ∋ x → ψ(x)(γ) ∈ R, γ ∈ Γ.. Observe that ψϕ = id and ϕψ = id. Thus, ϕ is a homeomorphism with ⊔ ϕ−1 = ψ, where ϕ(ℓfp (Γ )) ⊂ ℓf1 (Γ ) and ψ(ℓf1 (Γ )) ⊂ ℓfp (Γ ). ⊓. For each space X, we denote C(X) = C(X, R). The Banach space CB (X) is defined as follows: • CB (X) = f ∈ C(X) supx∈X |f (x)| < ∞ with the sup-norm. f = sup |f (x)|. x∈X. This sup-norm of CB (X) induces the uniform convergence topology. If X is discrete and infinite, then we have CB (X) = ℓ∞ (X), and so, in particular, CB (N) = ℓ∞ . When X is compact, CB (X) = C(X) and the topology induced by the norm coincides with the compact-open topology. The uniform convergence topology of C(X) is induced by the following metric: d(f, g) = sup min{|f (x) − g(x)|, 1}. x∈X. As can be easily observed, CB (X) is closed and open in C(X) under the uniform convergence topology. Note that CB (X) is a component of the space C(X) because CB (X) is path-connected as a normed linear space. Regarding C(X) as a subspace of the product space RX , we can introduce a topology on C(X), which is called the pointwise convergence topology. With respect to this topology, lim fn = f. n→∞. ⇔. lim fn (x) = f (x) for every x ∈ X.. n→∞. The space C(X) with the pointwise convergence topology is usually denoted by Cp (X). The space Cp (N) is simply the space of sequences s = RN . In this chapter, three topologies on C(X) have been considered — the compact-open topology, the uniform convergence topology, and the pointwise convergence topology. Among them, the uniform convergence topology is the finest and the pointwise convergence topology is the coarsest..

(43) 20. 0 Preliminaries. Notes for Preliminaries Theorem 0.2.5 is due to Mazur [Ma]. Zhongqiang Yang pointed out that Proposition 0.2.4 can be applied to show the continuity of ϕ and ψ in the proof of Theorem 0.2.5. Related to Mazur’s result, R.D. Anderson [An] proved that = Ê is homeomorphic to the Hilbert space ℓ2 . For an elementary proof, refer to [AB].. References [An] R.D. Anderson, Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 72 (1966), 515–519. [AB] R.D. Anderson and R.H. Bing, A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 74 (1968), 771–792. [Ma] S. Mazur, Une remarque sur l’homéomorphie des champs fonctionnels, Studia Math. 1 (1929), 83–85..

(44) 1 Metrization and Paracompact Spaces. In this chapter, we are mainly concerned with metrization and paracompact spaces. We also derive a number of properties of the products of compact spaces and perfect maps. Several metrization theorems are proved, and we characterize completely metrizable spaces. We will study several different characteristics of paracompact spaces that indicate, in many situations, the advantages of paracompactness. In particular, there exists a useful theorem showing that, if a paracompact space has a certain property locally, then it has the same property globally. Furthermore, paracompact spaces have partitions of unity, which is also a very useful property.. 1.1 Products of Compact Spaces and Perfect Maps In this section, we present a number of theorems regarding the products of compact spaces and compactifications. In addition, we introduce perfect maps. First, we present a proof of the Tychonoff Theorem. 1.1.1 Theorem (Tychonoff) The product space λ∈Λ Xλ of compact spaces Xλ , λ ∈ Λ, is compact. Proof. Let X = λ∈Λ Xλ . We may assume that Λ = (Λ, ) is a well-ordered set. For each μ ∈ Λ, let pμ : X → λμ Xλ and qμ : X → λ<μ Xλ be the projections. Let A be a collection of subsets of X with the finite intersection property −1 (f.i.p.). Using transfinite induction, we can find xλ ∈ X λ such that A|pλ (U ) has the f.i.p. for every neighborhood U of (xν )νλ in νλ Xν . Indeed, suppose that xλ ∈ Xλ , λ < μ, have been found, but there exists no xμ ∈ Xμ with the above property, i.e., any y ∈ Xμhas an open neighborhood Vy with an open neighborhood Uy of (xλ )λ<μ in λ<μ Xλ such that A|qμ−1 (Uy )∩pr−1 μ (Vy ) does not have the f.i.p. Because Xμ is compact, we have y1 , . . . , yn ∈ Xμ n n such that Xμ = i=1 Vyi . Since i=1 Uyi is a neighborhood of (xλ )λ<μ in.