Then,V = st(A,U) =
U[A] is an open neighborhood ofA. SinceU is locally finite, it follows that clV =
U[A]cl. Since eachU ∈ U[A] is contained in some Ua, it follows thatx∈clU, and hencex∈clV.
We now show that X is collectionwise normal. Let F be a discrete col-lection of closed sets in X. Since X is regular, each x ∈ X has an open neighborhood Vx in X such that cardF[clVx]1. Let U be a locally finite open refinement of{Vx|x∈X} ∈cov(X). For each F∈ F, we define
WF =X\ clU U ∈ U, F ∩clU =∅ .
Then, WF is open inX andF ⊂WF ⊂st(F,Ucl). Since cardF[clU]1 for each U ∈ U, it follows that st(F,Ucl)∩WF′ =∅ ifF′ =F ∈ F. Therefore, {WF |F ∈ F}is pairwise disjoint. ⊓⊔
F′ F
WF U
WF′
st(F,Ucl)
Fig. 1.6. The pairwise disjoint collection{WF |F ∈ F}
1.6.2 Lemma IfX is regular and each open cover of X has a locally finite refinement (consisting of arbitrary sets), then for any open coverU ofX there is a locally finite closed cover{FU |U ∈ U}ofX such thatFU ⊂U for each U ∈ U.
Proof. SinceX is regular, we have V ∈cov(X) such that Vcl≺ U. Let Abe a locally finite refinement ofV. There exists a function ϕ:A → U such that clA⊂ϕ(A) for each A∈ A. For eachU ∈ U, define
FU = clAA∈ϕ−1(U)
⊂U.
Since eachx∈X is contained in someA∈ AandA⊂Fϕ(A),{FU |U ∈ U}is a cover ofX. SinceAis locally finite, eachFUis closed inXand{FU |U ∈ U}
is locally finite. ⊓⊔
We have the following characterizations of paracompactness:
1.6.3 Theorem For a spaceX, the following conditions are equivalent:
1.6 Paracompactness and Local Properties 47 (a) X is paracompact;
(b) Each open cover ofX has an open∆-refinement;
(c) Each open cover ofX has an open star-refinement;
(d) X is regular and each open cover ofX has aσ-discrete open refinement;
(e) X is regular and each open cover ofX has a locally finite refinement.
Proof. (a)⇒(b): LetU ∈cov(X). From Lemma 1.6.2, it follows thatX has a locally finite closed cover{FU |U ∈ U}such that FU ⊂U for eachU ∈ U. For eachx∈X, define
Wx= U ∈ Ux∈FU
\ FU
U ∈ U, x∈FU .
Then, Wx is an open neighborhood ofx in X, hence W ={Wx |x∈X} ∈ cov(X). For each x ∈ X, choose U ∈ U so that x ∈ FU. If x ∈ Wy then y∈FU, which implies thatWy⊂U. Therefore, st(x,W)⊂U for eachx∈X, which means thatW is a∆-refinement of U.
(b)⇒(c): Due to Proposition 1.4.3, for U,V,W ∈cov(X), W∆≺ V≺ U ⇒ W∆ ≺ U∗ .
This gives (b)⇒(c).
(c)⇒(d): To prove the regularity ofX, let A⊂X be closed andx∈X\A.
Then, {X \A, X\ {x}} ∈ cov(X) has an open star-refinement W. Choose W ∈ W so that x∈ W. Then, st(W,W) ⊂X \A, i.e., W ∩st(A,W) = ∅. Hence,X is regular.
Next, we show that each U ∈ cov(X) has a σ-discrete open refinement.
We may assume thatU ={Uλ |λ∈Λ}, where Λ= (Λ,) is a well-ordered set. By condition (c), we have a sequence of open star-refinements:
U ≻ U∗ 1≻ U∗ 2≻ · · ·∗ . For each (λ, n)∈Λ×N, let
Uλ,n= U ∈ Un st(U,Un)⊂Uλ
⊂Uλ.
Then, we have
(∗) st(Uλ,n,Un+1)⊂Uλ,n+1 for each (λ, n)∈Λ×N.
Indeed, eachU ∈ Un+1[Uλ,n] meets someU′ ∈ Un such that st(U′,Un)⊂Uλ. SinceU ⊂st(U′,Un+1), it follows that
st(U,Un+1)⊂st2(U′,Un+1)⊂st(U′,stUn+1)⊂st(U′,Un)⊂Uλ, which implies thatU ⊂Uλ,n+1. Thus, we have (∗).
Now, for each (λ, n)∈Λ×N, let
Vλ,n=Uλ,n\cl
μ<λUμ,n+1⊂Uλ.
Then, each Vn = {Vλ,n | λ ∈ Λ} is discrete in X. Indeed, each x ∈ X is contained in some U ∈ Un+1. If U ∩Vμ,n = ∅, then U ⊂ st(Uμ,n,Un+1) ⊂ Uμ,n+1 by (∗). Hence,U∩Vλ,n =∅ for allλ > μ. This implies that U meets at most one member ofVn.
µ<λ
Vµ,n+1
µ<λ
Uλ,n
Uλ
Uλ,n+1
Uλ,n
Uµ,n
Uµ,n+1
Vµ,n
Vλ,n
Fig. 1.7.Definition ofVλ,n
It remains to be proved that V =
n∈NVn ∈ cov(X). Each x ∈ X is contained in some U ∈ U1. Since st(U,U1)⊂ Uλ for some λ ∈ Λ, it follows that x∈Uλ,1. Thus, we can define
λ(x) = min
λ∈Λx∈Uλ,n for somen∈N . Then,x∈Uλ(x),nfor somen∈N. It follows from (∗) that
cl
μ<λ(x)Uμ,n+1⊂st μ<λ(x)Uμ,n+1,Un+2
=
μ<λ(x)
st(Uμ,n+1,Un+2)⊂
μ<λ(x)
Uμ,n+2,
hencex∈cl
μ<λ(x)Uμ,n+1. Therefore,x∈Vλ(x),n, and henceV ∈ cov(X).
Consequently,V is aσ-discrete open refinement ofU.
(d)⇒(e): It suffices to show that everyσ-discrete open coverU ofX has a locally finite refinement. Let U =
n∈NUn, where each Un is discrete inX and Un∩ Um =∅ ifn = m. For eachU ∈ Un, let AU =U \
m<n( Um).
Then, A = {AU | U ∈ U} is a cover of X that refines U. For each x ∈ X, choose the smallestn∈Nsuch thatx∈
Un and letx∈U0∈ Un. Then,U0
missesAU for allU ∈
m>nUm. For eachmn, sinceUmis discrete,xhas a neighborhoodVminX such that cardUm[Vm]1. Then,V =U0∩V1∩· · ·∩Vn
is a neighborhood ofxinXsuch that cardA[V]n. Hence,Ais locally finite in X.
(e)⇒(a): LetU ∈cov(X). ThenU has a locally finite refinementA. For eachx∈X, choose an open neighborhoodVxofxinX so that cardA[Vx]<
1.6 Paracompactness and Local Properties 49 ℵ0. According to Lemma 1.6.2, {Vx | x∈ X} ∈ cov(X) has a locally finite closed refinementF. Then, cardA[F]<ℵ0 for eachF ∈ F. For eachA∈ A, chooseUA∈ U so thatA⊂UAand define
WA=UA\ F ∈ F A∩F=∅ .
Then, A⊂WA⊂UAand WA is open inX, henceW ={WA|A∈ A}is an open refinement of U. Since F is a locally finite closed cover of X, st(x,F) is a neighborhood of x ∈ X. For each F ∈ F and A ∈ A, F ∩WA = ∅ impliesF∩A=∅. Then, cardW[F]cardA[F]<ℵ0for eachF ∈ F. Since cardF[x]<ℵ0, st(x,F) meets only finitely many members ofW. Hence,W is locally finite inX. ⊓⊔
x st(x,F)
UA
WA
A F
Fig. 1.8.Definition ofWA
A space X is Lindel¨of if every open cover of X has a countable open refinement. By verifying condition (d) above, we have the following:
1.6.4 Corollary Every regular Lindel¨of space is paracompact. ⊓⊔
LetPbe a property of subsets of a spaceX. It is said thatXhas property P locally if each x∈ X has a neighborhood U in X that has property P. Occasionally, we need to determine whetherX has some property P ifX has property P locally. Let us consider this problem now. A propertyP ofopen sets inX is said to beG-hereditaryif the following conditions are satisfied:
(G-1) IfU has propertyP, then every open subset ofU hasP; (G-2) IfU andV have propertyP, thenU∪V has propertyP;
(G-3) If{Uλ |λ∈Λ} is discrete inX and eachUλ has property P, then
λ∈ΛUλ has propertyP.
The following theorem is very useful to show that a space has a certain prop-erty:
1.6.5 Theorem (E. Michael) Let P be a G-hereditary property of open sets in a paracompact spaceX. IfX has propertyP locally, thenX itself has propertyP.
Proof. SinceXhas propertyPlocally, there existsU ∈cov(X) such that each U ∈ U has propertyP. According to Theorem 1.6.3,Uhas an open refinement V =
n∈NVn such that each Vn is discrete in X. EachV ∈ V has property P by (G-1). For eachn ∈N, letVn =
Vn. Then, eachVn has property P by (G-3), henceV1∪ · · · ∪Vn has propertyP by (G-2). From Lemma 1.6.2, it follows that X has a closed cover {Fn | n∈N} such that Fn ⊂Vn for each n∈N.4 Inductively choose open setsGn (n∈N) so that
Fn∪clGn−1⊂Gn ⊂clGn ⊂V1∪ · · · ∪Vn,
where G0 = ∅. For each n ∈ N, let Wn = Gn \clGn−2, where G−1 = ∅. Then, each Wn also has propertyP by (G-1). LetXi=
n∈ωW3n+i, where i= 1,2,3. Since{W3n+i |n∈ω}is discrete inX, eachXihas propertyP by (G-3). Hence,X=X1∪X2∪X3 also has propertyP by (G-2). ⊓⊔
V2
F3
V3
G1
F1 F2
V1
G3
G2
Fig. 1.9. Construction ofGn
There are many cases where we consider properties of closed sets rather than open sets. In such cases, Theorem 1.6.5 can also be applied. In fact, let P be a property ofclosed sets ofX. We define the property P◦ ofopen sets in X as follows:
U has propertyP◦⇐⇒
def clU has propertyP.
It is said thatP isF-hereditaryif it satisfies the following conditions:
(F-1) IfAhas propertyP, then every closed subset ofAhas propertyP; (F-2) IfAandB have propertyP, then A∪B has propertyP;
4Closed setsFn⊂X,n∈can be inductively obtained so thatX =
Ë
i nintFi∪
Ë
i>nVi.
1.6 Paracompactness and Local Properties 51 (F-3) If{Aλ |λ∈Λ} is discrete inX and each Aλ has property P, then
λ∈ΛAλ has propertyP.
Evidently, if propertyP is F-hereditary, then P◦ is G-hereditary. Therefore, Theorem 1.6.5 yields the following corollary:
1.6.6 Corollary (E. Michael) LetP be anF-hereditary property of closed sets in a paracompact spaceX. IfX has propertyP locally, thenX itself has propertyP. ⊓⊔
1.6.7 Additional Results on Paracompact Spaces
(1) A space is paracompact if it is a locally finite union of paracompact closed subspaces.
Sketch of Proof.LetFbe a locally finite closed cover of a spaceXsuch that each F ∈ F is paracompact. To prove regularity, letx∈ X and U an open neighborhood ofxinX. Since eachF∈ F[x] is regular, we have an open neighborhoodUF ofxinX such that cl(F∩UF) ⊂U. The followingU0 is an open neighborhood ofxinX:
U0=
F∈F[x]
UF\
(F \ F[x])
⊂
F[x] = st(x,F)
.
Observe that clXU0 = cl F∈F[x](U0∩F) = F∈F[x]cl(U0∩F)⊂U. Thus, it suffices to show thatX satisfies condition 1.6.3(e).
(2) EveryFσ subspace Aof a paracompact spaceX is paracompact.
Sketch of Proof.It suffices to show thatA satisfies condition 1.6.3(d).
LetA= n∈
An, where eachAnis closed inX. For eachV ∈cov(A) andn∈, let
Un=
X\An
∪
V
V ∈ V
∈cov(X),
where eachVis open inXwithV∩A=V. Note thatVn≺ Unimplies thatVn[An]|A≺ V.
(3) Let X be a paracompact space. If every open subspace ofX is paracom-pact, then every subspace ofX is also paracompact.
Sketch of Proof.To find a locally finite open refinement ofU ∈cov(A), take an open collectionUinX such thatU |A =U and use the para-compactness of U.
(4) A paracompact space X is (completely) metrizable if it is locally (com-pletely) metrizable.
Sketch of Proof.To apply 1.4.5(2) (1.5.10(2)), construct a locally finite cover ofX consisting of (completely) metrizable closed sets.
A spaceX ishereditarily paracompactif every subspace ofX is para-compact. The following theorem comes from (2) and (3).
1.6.8 Theorem Every perfectly normal paracompact space is hereditarily paracompact. ⊓⊔