(2) If bothY andZ are paracompact, the composition
C(X, Y)×C(Y, Z)∋(f, g)→g◦f ∈C(X, Z) is continuous with respect to the limitation topology.
Sketch of Proof.For each (f, g)∈C(X, Y)×C(Y, Z) andU ∈cov(Z), let V ∈cov(Z) be a star-refinement ofU. Show thatf′ ∈ g−1(V)(f) andg′∈ V(g) impliesg′◦f′∈ U(g◦f).
(3) For every paracompact spaceX, the inverse operation Homeo(X)∋h→h−1∈Homeo(X)
is continuous with respect to the limitation topology. Combining this with (1), the group Homeo(X) with the limitation topology is a topological group.
Sketch of Proof.Leth ∈Homeo(X) andU ∈cov(X). Show thatg ∈ h(U)(h) impliesg−1∈ U(h−1).
Remark 12.If Y = (Y, d) is a metric space, for each f ∈ C(X, Y) and γ ∈ C(X,(0,∞)), let
Vγ(f) =
g∈C(X, Y)∀x∈X, d(f(x), g(x))< γ(x) .
We have the topology of C(X, Y) such that {Vγ(f)|γ ∈C(X,(0,∞))} is a neighborhood basis off. This is finer than the limitation topology. In general, these topologies are not equal.
For example, letγ ∈C(N,(0,∞)) be the map defined byγ(n) = 2−n for n ∈N. Then, Vγ(0) is not a neighborhood of 0∈ C(N,R) in the limitation topology. Indeed, for any α∈C(R,(0,∞)), we define g∈C(N,R) byg(n) =
1
2α(0) for everyn∈N. Then,g∈Nα(0) butg∈Vγ(0). Thus,Nα(0)⊂Vγ(0).
Moreover, the composition
C(N,R)×C(R,R)∋(f, g)→g◦f ∈C(N,R) is not continuous with respect to this topology.
Indeed, letγ be the above map. For anyα∈C(Ê,(0,∞)), we haven∈Æ such that 2−n< 12α(0). Leth= id +12α∈C(Ê,Ê). Then,h∈Vα(id) but h◦0∈Vγ(id◦0) becauseh◦0(n) =h(0) = 12α(0)>2−n=γ(n). (Here, id can be replaced by anyg∈C(Ê,Ê).)
1.10 Counter-examples 67 1.10.1 The Tychonoff plank Let [0, ω1) be the space of all countable ordinals with the order topology. The space [0, ω1] is the one-point compacti-fication of the space [0, ω1). Let [0, ω] be the one-point compactification of the space ω = [0, ω) of non-negative integers. The product space [0, ω1]×[0, ω]
is a compact Hausdorff space, hence it is paracompact. The following dense subspace of [0, ω1]×[0, ω] is called the Tychonoff plank:
T = [0, ω1]×[0, ω]\ {(ω1, ω)}. We now prove that
− The Tychonoff plankT is not normal.
Proof. We have disjoint closed sets{ω1}×[0, ω) and [0, ω1)×{ω}inT. Assume that T has disjoint open sets U,V such that{ω1} ×[0, ω)⊂U and [0, ω1)× {ω} ⊂ V. For each n∈ ω, choose αn < ω1 so that [αn, ω1]× {n} ⊂U. Let α = supn∈Nαn < ω1. Then, [α, ω1]×N ⊂ U. On the other hand, we can choose n∈Nso that{α} ×[n, ω]⊂V. Thus, we haveU∩V =∅, which is a contradiction. ⊓⊔
[0, ω1] [0, ω]
ω
ω1
(ω1, ω)
[αn, ω1]× {n}
αn
α= supn∈αn< ω1
U V
◦ {α} ×[n, ω]
Fig. 1.10.Tychonoff plank
The next example shows thatthe concepts of normality, perfect normality, hereditary normality, collectionwise normality, and paracompactness are not productive.
1.10.2 The Sorgenfrey line TheSorgenfrey lineS is the spaceRwith the topology generated by [a, b),a < b. The productS2is called the Sorgen-frey plane. These spaces have the following properties:
(1) Sis a separable regular Lindel¨of space, hence it is paracompact, and so is collectionwise normal;
(2) S is perfectly normal, and so is hereditarily normal;
(3) S2is not normal.
Proof. (1): It is obvious thatS is Hausdorff. Since each basic open set [a, b) is also closed in S, it follows that S is regular. Clearly, Q is dense in S, hence S is separable. To see that S is Lindel¨of, let U ∈ cov(S). We have a function γ : S → Q so that γ(x) > x and [x, γ(x)) ⊂ U for some U ∈ U. Then, {[x, γ(x)) | x ∈ S} ∈ cov(S) is an open refinement of U. For each q ∈ γ(S), if there exists minγ−1(q), let R(q) = {minγ−1(q)}. Otherwise, choose a countable subsetR(q)⊂γ−1(q) so that infR(q) = infγ−1(q), where we meanγ−1(q) =−∞ifγ−1(q) is unbounded below. Then, the following is a subcover of{[x, γ(x))|x∈S} ∈cov(S):
[z, q)q∈γ(S), z∈R(q)
∈cov(S), which is a countable open refinement ofU.
(2): Let U be an open set in S. We have a function γ : U → Q so that γ(x)> xand [x, γ(x))⊂U. Then,U =
x∈U[x, γ(x)). By the same argument as the proof of (1), we can find a countable subcollection
[ai, bi)i∈N
⊂
[x, γ(x))x∈U such thatU =
i∈N[ai, bi), hence U isFσ inS. Thus,S is perfectly normal.
(3): As we saw in the proof of (1),Qis dense in S, henceQ2 is dense in S2. It follows that the restriction C(S2,R)∋ f → f|Q2 ∈ RQ2 is injective.
Therefore,
card C(S2,R)cardRQ2= 2ℵ0 =c.
On the other hand, D = {(x, y) ∈ S2 | x+y = 0} is a discrete set in S2. Then, we have
card C(D,R) = cardRD= 2c>ccard C(S2,R).
If S2 is normal, it would follow from the Tietze Extension Theorem 1.2.2 that the restriction C(S2,R)∋f → f|D ∈C(D,R) is surjective, which is a contradiction. Consequently,S2 is not normal. ⊓⊔
Finally, we will construct a closed tower such that the direct limit isnot Hausdorff.
1.10.3 A non-Hausdorff direct limit LetY be a space which is Hausdorff but non-normal, such as the Tychonoff plank. Let A0, A1 be disjoint closed sets inY that have no disjoint neighborhoods. We defineX = (Y×N)∪{0,1} with the topology generated by open sets in the product spaceY×Nand sets
of the form
k>n
(Uk× {k})∪ {i},
where i = 0,1 and eachUk is an open neighborhood ofAi. Then, X is not Hausdorff because 0 and 1 have no disjoint neighborhoods in X. For each n∈N, let
Notes for Chapter 1 69
1 2 3
A0 A1
0 1
Y
X1 X2 X = (Y ×)∪ {0,1}
0 1
Fig. 1.11.Non-Hausdorff direct limit
Xn =Y × {1, . . . , n} ∪(A0∪A1)× {k|k > n} ∪ {0,1}. Then, X1 ⊂ X2 ⊂ · · · are closed in X and X =
n∈NXn. As is easily observed, everyXn is Hausdorff. We will prove thatX= lim
−→Xn, that is,
− X has the weak topology with respect to the tower (Xn)n∈N.
Proof. Since id : lim−→Xn →X is obviously continuous, it suffices to show that every open setV in lim−→Xn is open inX. To this end, assume thatV ∩Xn is open inXnfor eachn∈N. Eachx∈V\{0,1}is contained in someY×{n} ⊂ Xn. Then,V∩(Y×{n}) is an open neighborhood ofxinY×{n}, and so is an open neighborhood inX. When 0∈V,A0× {k|k > n} ⊂V for somen∈N becauseV ∩X1is open inX1. For eachk > n, sinceV∩(Y× {k}) is open in Y×{k}, there is an open setUkinY such thatV∩(Y×{k}) =Uk×{k}. Note that A0⊂Uk. Then,
k>n(Uk× {k})∪ {0} ⊂V, henceV is a neighborhood of 0 inX. Similarly,V is a neighborhood of 1 inX if 1∈V. Thus,V is open in X. ⊓⊔
Notes for Chapter 1
For more comprehensive studies on General Topology, see Engelking’s book, which contains excellent historical and bibliographic notes at the end of each section.
• R. Engelking, General Topology, Revised and complete edition, Sigma Ser. in Pure Math.6, Heldermann Verlag, Berlin, 1989.
The following classical books are still good sources.
• J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, 1966.
• J.L. Kelly, General Topology, GTM27, Springer-Verlag, Berlin, (Reprint of the 1955 ed. published by Van Nostrand).
For counter-examples, the following is a good reference:
• L.A. Steen and J.A. Seebach, Jr., Counterexamples in Topology (2nd edition), Springer-Verlag, New York, 1978.
Of the more recent publications, the following textbook is readable and seems to be popular:
• J.R. Munkres, Topology (2nd edition), Prentice Hall, Inc., Upper Saddle River, 2000.
Most of the contents discussed in the present chapter are found in Chapters 5–8 of this text, although it does not discuss the Frink Metrization Theorem (cf. 1.4.1) and Michael’s Theorem 1.6.5 on local properties.
Among various proofs of the Tychonoff Theorem 1.1.1, our proof is a modifica-tion of the proof due to D.G. Wright [Wr]. Our proof of the Tietze Extension The-orem 1.2.2 is due to B.M. Scott [Sc]. TheThe-orem 1.3.1 was established by A.H. Stone [St], but the proof presented here is due to M.E. Rudin [Ru]. The Nagata–Smirnov Metrization Theorem (cf. 1.3.4) was independently proved by J. Nagata [Na] and Ju.M. Smirnov [Sm]. The Bing Metrization Theorem (cf. metrization) was proved in [Bi]. The Urysohn Metrization Theorem 1.3.5 and the Alexandroff–Urysohn Metriza-tion Theorem (cf. 1.4.1) were established in [Ur] and [AU], respectively. The Frink Metrization Theorem (cf. 2nd-metrization) was proved by A.H. Frink [Fr]. The Baire Category Theorem 1.5.1 was first proved by Hausdorff [Ha] (Baire proved the theo-rem for the real line in 1889). The equivalence of (a) and (b) in Theotheo-rem 1.5.5 was shown by ˇCech [ ˇCe]. Theorems 1.5.7 and 1.5.8 were established by Lavrentieff [La].
The concept of paracompactness was introduced by J. Dieudonn´e [Di]. In [Bi], R.H. Bing introduced the concept of collectionwise normality and showed the collec-tionwise normality of paracompact spaces (Theorem 1.6.1). The equivalence of (b) and (c) in Theorem 1.6.3 was proved by J.W. Tukey [Tu], where he called spaces satisfying condition (c)fully normal spaces. The equivalence of (a) and (c) and the equivalence of (a), (d), and (e) were respectively proved by A.H. Stone [St] and E. Michael [Mi1]. Theorem 1.6.5 on local properties was established by E. Michael [Mi2]. Lemma 1.7.1 appeared in [Le]. Theorem 1.7.2 and Proposition 1.7.4 were also established by E. Michael [Mi2]. The simple proof of Proposition 1.7.4 presented here is due to M.R. Mather [Ma]. Theorem 1.7.6 was proved by Dieudonn´e [Di].
These notes are based on historical and bibliographic notes in Engelking’s book, listed above.
In some literature, it is mentioned that the direct limit of a closed tower of Hausdorff spaces need not be Hausdorff. The author could not find such an example in the literature. Example 1.10.3 is due to H. Ohta.
References
[AU] P. Alexandroff and P. Urysohn, Une condition n´ecessaire et suffisante pour qu’une classe(L) soit une classe(B),C.R. Acad. Sci. Paris S´er.
A-B177(1923), 1274–1276.
[Bi] R.H. Bing,Metrization of topological spaces,Canad. J. Math.3(1951), 175–186.
[ ˇCe] E. ˇCech,On bicompact spaces,Ann. Math.38(1937), 823–844.
[Di] J. Dieudonn´e,Une g´en´eralisation des espaces compacts,J. Math. Pures et Appl.23(1944), 65–76.
Notes for Chapter 1 71 [Fr] A.H. Frink, Distance functions and the metrization problems, Bull.
Amer. Math. Soc.43(1937), 133–142.
[Ha] F. Hausdorff, Grundz¨uge der Mengenlehre, Teubner, Leipzig, 1914.
[La] M. Lavrentieff,Contribution ´a la th´eorie des ensembles hom´eomorphes, Fund. Math.6(1924), 149–160.
[Le] S. Lefschetz, Algebraic Topology, Amer. Math. Soc. Colloq. Publ.27, Amer. Math. Soc., New York, 1942.
[Ma] M.R. Mather, Paracompactness and partition of unity, Ph.D. thesis, Cambridge Univ., 1965.
[Mi1] E. Michael,A note on paracompact spaces,Proc. Amer. Math. Soc.4 (1953), 831–838.
[Mi2] , Local properties of topological spaces, Duke Math. J. 21 (1954), 163–171.
[Na] J. Nagata,On a necessary and sufficient condition of metrizability,J.
Inst. Polyt. Osaka City Univ.1(1950), 93–100.
[Ru] M.E. Rudin, A new proof that metric spaces are paracompact, Proc.
Amer. Math. Soc.20(1969), 603.
[Sc] B.M. Scott,A “more topological” proof of the Tietze-Urysohn Theorem, Amer. Math. Monthly85(1978), 192–193.
[Sm] Ju.M. Smirnov,On the metrization of topological spaces(Russian), Us-pekhi Mat. Nauk6(no. 6) (1951), 100–111.
[St] A.H. Stone,Paracompactness and product spaces, Bull. Amer. Math.
Soc.54(1948), 977–982.
[Tu] J.W. Tukey, Convergence and uniformity in topology, Ann. Math. Stud-ies2, Princeton Univ. Press, Princeton, 1940.
[Ur] P. Urysohn,Uber die Metrisation der kompakten topologischen R¨¨ aume, Math. Ann.92(1924), 275–293.
[Wr] D.G. Wright,Tychonoff’s theorem,Proc. Amer. Math. Soc.120(1994), 985–987.
2
Topology of Linear Spaces and Convex Sets
In this chapter, several basic results on topological linear spaces and convex sets are presented. We will characterize finite-dimensionality, metrizability, and normability of topological linear spaces. Among the important results are the Hahn–Banach Extension Theorem, the Separation Theorem, the Closed Graph Theorem, and the Open Mapping Theorem. We will also prove the Michael Selection Theorem, which will be applied in the proof of the Bartle-Graves Theorem.