Vn≺ Un and Vn
≺ V∆ n−1
Vn
≺ V∗ n−1
,
where V0 =U. Letd′ ∈Metr(X) be the bounded metric obtained by applying Corollary 1.4.2 (or 1.4.4) with Remark 3 (or 4). For a given d∈Metr(X), the desiredρ∈Metr(X) can be defined by ρ= 8d′+d (orρ= 2d′+d).
(2) LetX = (X, d) be a metric space. For each open coverU ofX, there is a mapγ:X →(0,1) such that
B(x, γ(x))x∈X
≺ U. Sketch of Proof.For eachx∈X, let
r(x) = sup
U∈Umin{1, d(x, X\U)}= sup
U∈U
d(x, X¯ \U),
where ¯d = min{1, d}. Show that r : X → (0,∞) is l.s.c. Then, we can apply Theorem 1.7.6 to obtain a map γ : X → (0,1) such that γ(x)< r(x) for eachx∈X.
Remark.IfUis locally finite,ris continuous (in fact,ris 1-Lipschitz), so we can defineγ= 12r.
1.8 The Direct Limits of Towers of Spaces 57 (intR∞W)∩Rn ⊂intRn(W∩Rn) = (−2−n,2−n)n for eachn∈N.
Then, it follows that (intR∞W)∩R⊂
n∈N(−2−n,2−n) ={0}, which means that (intR∞W)∩R=∅, and hence 0∈intR∞W.
It should also be noted that the direct limit lim−→Xn isT1 but, in general, non-Hausdorff. Such an example is shown in 1.10.3.
As is easily observed, lim
−→Xn(i)= lim
−→Xn for anyn(1)< n(2)<· · · ∈N. It is also easy to prove the following proposition:
1.8.1 Proposition Let X1 ⊂ X2 ⊂ · · · and Y1 ⊂ Y2 ⊂ · · · be towers of spaces. Suppose that there existn(1)< n(2)<· · ·,m(1) < m(2)<· · · ∈ N and mapsfi:Xn(i)→Ym(i)andgi:Ym(i)→Xn(i+1)such thatgifi= idXn(i)
andfi+1gi= idYm(i), that is, the following diagram is commutative:
Xn(1) f1
⊂ Xn(2) f2
⊂ Xn(3) f3
⊂ · · ·
Ym(1) g1
;;w
ww ww ww ww
⊂ Ym(2) g2
;;w
ww ww ww ww
⊂ Ym(3) ⊂ · · · Then,lim
−→Xn is homeomorphic tolim
−→Yn. ⊓⊔ Remark 7.It should be noted that lim
−→Xn is not a subspace of lim
−→Yn even if eachXn is a closed subspace ofYn. For example, letYn =Rbe the real line and
Xn ={0} ∪[n−1,1]⊂Yn =R.
Then, I=
n∈NXn, R= lim−→Yn, and 0 is an isolated point of lim−→Xn but is not in the subspaceI⊂R.
On the other hand, as is easily observed, if eachXn is an open subspace ofYn then lim
−→Xn is an open subspace of lim
−→Yn. The following proposition is also rather obvious:
1.8.2 Proposition LetY1⊂Y2⊂ · · · be a tower of spaces. If X is a closed (resp. open) subspace of Y = lim
−→Yn, then X= lim
−→(X∩Yn). Equivalently, if eachX∩Yn is closed (resp. open) inYn, thenlim
−→(X∩Yn)is a closed (resp.
open) subspace ofY. ⊓⊔
Remark 8.In general, X = lim
−→(X ∩Yn) for a subspace X ⊂ lim
−→Yn. For example, letYn be a subspace of the Euclidean planeR2 defined by
Yn=
(0,0), (i−1,0), (j−1, k−1)i, k∈N, j= 1, . . . , n .
Observe that A ={(j−1, k−1)| j, k ∈N} is dense in lim−→Yn, hence it is not closed in the following subspaceX of lim−→Yn:
X ={(0,0)} ∪ {(j−1, k−1)|j, k∈N}, whereasAis closed in lim−→(X∩Yn).
With regard to products of direct limits, we have:
1.8.3 Proposition LetX1⊂X2⊂ · · · be a tower of spaces. If Y is locally compact then(lim−→Xn)×Y = lim−→(Xn×Y)as spaces.
Proof. First of all, note that
(lim−→Xn)×Y = lim−→(Xn×Y) =
n∈N
(Xn×Y) as sets.
It is easy to see that id : lim−→(Xn×Y)→(lim−→Xn)×Y is continuous. To see this is an open map, letW be an open set in lim
−→(Xn×Y). For each (x, y)∈W, choose m∈N so thatx∈Xm. SinceY is locally compact, there exist open sets Um⊂Xm and V ⊂Y such thatx∈Um,y ∈V, Um×clY V ⊂W and clY V is compact. Then, by the compactness of clY V, we can find an open set Um+1⊂Xm+1such thatUm⊂Um+1andUm+1×clY V ⊂W. Inductively, we can obtain Um⊂Um+1 ⊂Um+2 ⊂ · · · such that each Un is open inXn and Un×clY V ⊂W. Then,U =
nmUn is open in lim−→Xn, and henceU×V is an open neighborhood of (x, y) in (lim
−→Xn)×Y with U×V ⊂W. Thus,W is open in (lim−→Xn)×Y. ⊓⊔
1.8.4 Proposition Let X1 ⊂ X2 ⊂ · · · and Y1 ⊂ Y2 ⊂ · · · be towers of spaces. If eachXn andYn are locally compact, then
lim−→Xn×lim−→Yn= lim−→(Xn×Yn) as spaces.
Proof. First of all, note that lim−→Xn×lim
−→Yn = lim
−→(Xn×Yn) =
n∈N
(Xn×Yn) as sets.
It is easy to see that id : lim
−→(Xn×Yn) → lim
−→Xn×lim
−→Yn is continuous.
To see that this is open, let W be an open set in lim−→(Xn ×Yn). For each (x, y)∈W, choose m∈N so that (x, y)∈ Xm×Ym. Since Xm and Ym are locally compact, we have open setsUm⊂XmandVm⊂Ymsuch that
x∈Um, y∈Vm, clXmUm×clYmVm⊂W
and both clXmUm and clYmVm are compact. Then, by the compactness of clXmUmand clYmVm, we can easily find open setsUm+1⊂Xm+1andVm+1 ⊂ Ym+1 such that
clXmUm⊂Um+1, clYmVm⊂Vm+1, clXm+1Um+1×clYm+1Vm+1⊂W and both clXm+1Um+1 and clYm+1Vm+1 are compact. Inductively, we can obtain Um ⊂ Um+1 ⊂ Um+2 ⊂ · · · and Vm ⊂ Vm+1 ⊂ Vm+2 ⊂ · · · such that Un and Vn are open in Xn and Yn, respectively, clXnUn and clYnVn
are compact, and clXnUn ×clYnVn ⊂ W. Then, U =
nmUn and V =
nmVnare open in lim
−→Xnand lim
−→Yn, respectively, and (x, y)∈U×V ⊂W. Therefore,W is open in lim−→Xn×lim−→Yn. ⊓⊔
1.8 The Direct Limits of Towers of Spaces 59 A tower X1 ⊂ X2 ⊂ · · · of spaces is said to be closed if each Xn is closed inXn+1; equivalently, eachXnis closed in the direct limit lim−→Xn. For a pointed space X= (X,∗), let
XfN=
x∈XNx(n) =∗ except for finitely manyn∈N
⊂XN. Identifying each Xn with Xn× {(∗,∗, . . .)} ⊂ XfN, we have a closed tower X ⊂X2⊂X3⊂ · · · withXfN=
n∈NXn. We writeX∞= lim
−→Xn, which is the spaceXfN with the weak topology with respect to the tower (Xn)n∈N. A typical example isR∞, which appeared in Remark 6.
1.8.5 Proposition Let X = (X,∗) be a pointed locally compact space.
Then, each x ∈ X∞ = lim−→Xn has a neighborhood basis consisting of X∞∩
n∈NVn, where eachVn is a neighborhood ofx(n)in Xn.5 Sketch of Proof.LetUbe an open neighborhood ofxinX∞. Choosen0∈ so thatx∈Xn0. For each i= 1, . . . , n0, eachx(i) has a neighborhoodVi
inX such that clViis compact and
n0
i=1clVi⊂U∩Xn0. Recall that we identifyXn−1=Xn−1×{∗} ⊂Xn. Forn > n0, we can inductively choose a neighborhoodVnofx(n) =∗inXso that clVnis compact and
n
i=1clVi⊂ U∩Xn, where we use the compactness ofn−1i=1 clVi
=n−1i=1 clVi× {∗}
.
— This is an excellent exercise as the first part of the proof of Wallace’s Theorem 1.1.2.
Remark 9.Proposition 1.8.3 does not hold without the local compactness ofY even if eachXnis locally compact. For example, (lim
−→Rn)×ℓ2= lim
−→(Rn×ℓ2).
Indeed, each Rn is identified with Rn× {0} ⊂RNf ⊂ℓ2. Then, we regard (lim−→Rn)×ℓ2= lim
−→(Rn×ℓ2) =RNf×ℓ2 as sets.
Consider the following set:
D=
(k−1en, n−1ek)∈RNf×ℓ2
k, n∈N ,
where eachei∈RNf ⊂ℓ2is the unit vector defined by ei(i) = 1 andei(j) = 0 forj=i. For eachn∈N, let
Dn=
(k−1en, n−1ek)k∈N .
Since {n−1ek | k ∈ N} is discrete in ℓ2, it follows that Dn is discrete (so closed) inRn×ℓ2, hence it is also closed inRm×ℓ2for everymn. Observe that D∩(Rn×ℓ2) = n
i=1Di. Then, D is closed in lim
−→(Rn×ℓ2). On the other hand, for each neighborhoodU of (0,0) in (lim−→Rn)×ℓ2, we can apply Proposition 1.8.5 to takeδi>0 (i∈N) andn∈Nso that
5In other words, the topology of lim−→Xnis a relative (subspace) topology inherited from the box topology ofX.
RNf∩
i∈N
[−δi, δi]
×n−1Bℓ2⊂U,
where Bℓ2 is the unit closed ball of ℓ2. Choose k ∈ N so that k−1 < δn. Then, (k−1en, n−1ek)∈U, which impliesU∩D =∅. Thus,D is not closed in (lim
−→Rn)×ℓ2.
Remark 10.In Proposition 1.8.4, it is necessary to assume that bothXn and Yn are locally compact. Indeed, letXn = Rn and Yn = ℓ2 for every n ∈N.
Then, lim−→Xn ×lim−→Yn = lim−→(Xn ×Yn), as we saw in the above remark.
Furthermore, this equality does not hold even if Xn = Yn. For example, lim−→(ℓ2)n×lim
−→(ℓ2)n= lim
−→((ℓ2)n×(ℓ2)n). Indeed, consider lim−→(ℓ2)n×lim
−→(ℓ2)n= lim
−→((ℓ2)n×(ℓ2)n) = (ℓ2)Nf ×(ℓ2)Nf as sets.
Identifying Rn = (Re1)n ⊂ (ℓ2)n and ℓ2 = ℓ2× {0} ⊂ (ℓ2)Nf, we can also consider
(lim−→Rn)×ℓ2= lim
−→(Rn×ℓ2) =RNf×ℓ2⊂(ℓ2)Nf ×(ℓ2)Nf as sets.
By Proposition 1.8.2, (lim
−→Rn)×ℓ2 and lim
−→(Rn×ℓ2) are closed subspaces of lim
−→(ℓ2)n×lim
−→(ℓ2)n and lim
−→((ℓ2)n×(ℓ2)n), respectively. As we saw above, (lim−→Rn)×ℓ2= lim−→(Rn×ℓ2). Thus, lim−→(ℓ2)n×lim−→(ℓ2)n = lim−→((ℓ2)n×(ℓ2)n).
1.8.6 Theorem For the direct limitX = lim−→Xn of a towerX1⊂X2⊂ · · · of spaces, the following hold:
(1) Every compact setA⊂X is contained in someXn.
(2) For each mapf :Y →X from a first countable spaceY toX, each point y∈Y has a neighborhoodV in Y such that the imagef(V)is contained in someXn. In particular, if A ⊂X is a metrizable subspace then each point ofAhas a neighborhood inAthat is contained in someXn. Proof. (1): Assume that Ais not contained in anyXn. For eachn∈N, take xn∈A\Xn and letD={xn |n∈N} ⊂A. Then,D is infinite and discrete in lim
−→Xn. Indeed, everyC ⊂D is closed in lim
−→Xn because C∩Xn is finite for eachn∈N. This contradicts the compactness ofA.
(2): Let {Vn | n ∈ N} be a neighborhood basis of y0 in Y such that Vn ⊂ Vn−1. Assume that f(Vn) ⊂ Xn for everyn ∈ N. Then, taking yn ∈ Vn\f−1(Xn), we have a compact setA={yn |n∈ω}inY. Due to (1),f(A) is contained in some Xm, and hence f(ym)∈Xm. Thich is a contradiction.
Therefore,f(Vn)⊂Xn for somen∈N. ⊓⊔
By Theorem 1.8.6(2), the direct limit of metrizable spaces is non-metrizable in general (e.g., lim−→Rn is non-metrizable). However, it has some favorable properties, which we now discuss.
1.8 The Direct Limits of Towers of Spaces 61 1.8.7 Theorem For the direct limit X = lim−→Xn of a closed tower X1 ⊂ X2⊂ · · · of spaces, the following properties hold:
(1) If eachXn is normal, thenX is also normal;
(2) If eachXn is perfectly normal, thenX is also perfectly normal;
(3) If eachXn is collectionwise normal, thenX is also collectionwise normal;
(4) If eachXn is paracompact, thenX is also paracompact.
Proof. (1): Obviously, every singleton ofX is closed, so X is T1. LetA and B be disjoint closed sets in X. Then, we have a mapf1 :X1 →Isuch that f1(A∩X1) = 0 and f1(B∩X1) = 1. Using the Tietze Extension Theorem 1.2.2, we can extendf1 to a mapf2 :X2→Isuch that f2(A∩X2) = 0 and f2(B∩X2) = 1. Thus, we inductively obtain mapsfn:Xn →I,n∈N, such that
fn|Xn−1=fn−1, fn(A∩Xn) = 0 and fn(B∩Xn) = 1.
Let f : X → I be the map defined by f|Xn = fn for n ∈ N. Evidently, f(A) = 0 andf(B) = 1. Therefore,X is normal.
(2): From (1), it suffices to show that every closed setA inX is a Gδ set.
Each Xn has open sets Gn,m, m∈N, such thatA∩Xn =
m∈NGn,m. For each n, m∈N, letG∗n,m=Gn,m∪(X\Xn). SinceXn is closed in X, each G∗n,mis open inX. Observe thatA=
n,m∈NG∗n,m. Hence,AisGδ in X.
(3): Let F be a discrete collection of closed sets in X. By induction on n ∈ N, we have discrete collections{UnF | F ∈ F}of open sets in Xn such that (F ∩Xn)∪clUnF−1 ⊂ UnF for each F ∈ F, where U0F = ∅. For each F ∈ F, let UF =
n∈NUnF. Then, F ⊂ UF and UF is open in X because UF ∩Xn =
inUiF ∩Xn is open in Xn for eachn ∈ N. If F = F′, then UF ∩UF′ =∅because
UiF∩UjF′ ⊂UmaxF {i,j}∩UmaxF′ {i,j}=∅ for eachi, j∈N. Therefore,X is collectionwise normal.
(4): Since every paracompact space is collectionwise normal (Theorem 1.6.1), X is also collectionwise normal by (3), so it is regular. Then, due to Theorem 1.6.3, it suffices to show that each U ∈cov(X) has aσ-discrete open refinement. By Theorem 1.6.3, we have
m∈NVn,m∈ cov(Xn),n∈N, such that each Vn,m is discrete in Xn and Vn,m ≺ U. For each V ∈ Vn,m, choose UV ∈ U so that V ⊂UV. Note that eachVn,mcl is discrete in X, and recall that X is collectionwise normal. So, X has a discrete open collection {WV |V ∈ Vn,m} such that clV ⊂WV. LetWn,m={WV ∩UV |V ∈ Vn,m}. Then, W =
n,m∈NWn,m∈cov(X) is aσ-discrete open cover refinement of U. ⊓⊔
From Theorems 1.8.7 and 1.6.8, we conclude the following:
1.8.8 Corollary The direct limit of a closed tower of metrizable spaces is perfectly normal and paracompact, and so it is hereditarily paracompact. ⊓⊔