In mathematics, Michael's theorem gives sufficient conditions for a regular topological space (in fact, for a T1-space) to be paracompact.
Statement A family E i {\displaystyle E_{i}} of subsets of a topological space is said to be closure-preserving if for every subfamily E i j {\displaystyle E_{i_{j}}} ,
⋃ E i j ¯ = ⋃ E i j ¯ {\displaystyle {\overline {\bigcup E_{i_{j}}}}=\bigcup {\overline {E_{i_{j}}}}} . For example, a locally finite family of subsets has this property. With this terminology, the theorem states:
Frequently, the theorem is stated in the following form:
In particular, a regular-Hausdorff Lindelöf space is paracompact. The proof of the theorem uses the following result which does not need regularity:
Proof of Theorem from Proposition Here is a proof sketch of the theorem from the proposition. (1) ⇒ {\displaystyle \Rightarrow } (2) trivial. (2) ⇒ {\displaystyle \Rightarrow } (3) Since X {\displaystyle X} is regular, we can find an open cover V = { V x ∣ x ∈ X } {\displaystyle {\mathcal {V}}=\{V_{x}\mid x\in X\}} such that { V x ¯ ∣ x ∈ X } {\displaystyle \{{\overline {V_{x}}}\mid x\in X\}} is a refinement of a given open cover U {\displaystyle {\mathcal {U}}} . By (2), V {\displaystyle {\mathcal {V}}} has a closure-preserving refinement C {\displaystyle {\mathcal {C}}} . Then { C ¯ ∣ C ∈ C } {\displaystyle \{{\overline {C}}\mid C\in {\mathcal {C}}\}} is a refinement of U {\displaystyle {\mathcal {U}}} and is clearly closure-preserving. (3) ⇒ {\displaystyle \Rightarrow } (4) by Proposition, and (4) ⇒ {\displaystyle \Rightarrow } (1) again by Proposition (or rather the argument used in the proof of Proposition). ◻ {\displaystyle \square }
Examples Here are some examples of refinements satisfying the condition of the corollary.
A countable open refinement trivially satisfies the condition of the corollary. Thus, a regular (Hausdorff) Lindelöf space is paracompact. Let X {\displaystyle X} be a metric space and U a , a ∈ A {\displaystyle U_{a},a\in A} an open cover indexed by a well-ordered set A {\displaystyle A} . Then let
U a , n = { x ∣ d ( x , X − U a ) > 2 − n } , {\displaystyle U_{a,n}=\{x\mid d(x,X-U_{a})>2^{-n}\},}
T a , n = U a , n ∩ ( ∩ b < a ( X − U b ) ) . {\displaystyle T_{a,n}=U_{a,n}\cap (\cap _{b<a}(X-U_{b})).}
Then d ( T a , n , T b , n ) ≥ 2 − n {\displaystyle d(T_{a,n},T_{b,n})\geq 2^{-n}} for b ≠ a {\displaystyle b\neq a} and, from that, for each n {\displaystyle n} , one can choose neighborhoods V a , n ⊃ T a , n {\displaystyle V_{a,n}\supset T_{a,n}} inside U a , n {\displaystyle U_{a,n}} such that { V a , n ∣ a } {\displaystyle \{V_{a,n}\mid a\}} is locally finite. Then { V a , n ∣ a , n } {\displaystyle \{V_{a,n}\mid a,n\}} is a refinement satisfying the condition of the corollary, and we conclude X {\displaystyle X} is paracompact.
Proof
Lemmas The proof of the proposition uses the following general lemmas:
Proof: Given an open cover U {\displaystyle {\mathcal {U}}} , let A {\displaystyle {\mathcal {A}}} be a locally finite closed refinement of it. By "locally finite", we can find an open cover { V x ∣ x ∈ X } {\displaystyle \{V_{x}\mid x\in X\}} of X {\displaystyle X} such that each V x {\displaystyle V_{x}} intersects only finitely many sets in A {\displaystyle {\mathcal {A}}} . Then let F {\displaystyle {\mathcal {F}}} be a locally finite closed refinement of { V x ∣ x } {\displaystyle \{V_{x}\mid x\}} . Then, for each A ∈ A {\displaystyle A\in {\mathcal {A}}} ,
W ( A ) := ∩ { X − F ∣ F ∈ F , F ∩ A = ∅ } {\displaystyle W(A):=\cap \{X-F\mid F\in {\mathcal {F}},\,F\cap A=\emptyset \}}
is an open set since F {\displaystyle {\mathcal {F}}} is locally finite; thus, closure-preserving. Let U : A → U {\displaystyle U:{\mathcal {A}}\to {\mathcal {U}}} be a function such that A ⊂ U ( A ) {\displaystyle A\subset U(A)} and then let
A ′ = U ( A ) ∩ W ( A ) , {\displaystyle A'=U(A)\cap W(A),}
which is open. Then { A ′ ∣ A ∈ A } {\displaystyle \{A'\mid A\in {\mathcal {A}}\}} is a locally finite open refinement of U {\displaystyle {\mathcal {U}}} (use F {\displaystyle {\mathcal {F}}} is a locally finite cover). ◻ {\displaystyle \square }
Proof: Let U = ∪ 1 ∞ U n {\displaystyle {\mathcal {U}}=\cup _{1}^{\infty }{\mathcal {U}}_{n}} be an open cover that is a countable union of locally finite collections U n {\displaystyle {\mathcal {U}}_{n}} . Then let U n = ∪ U n {\displaystyle U_{n}=\cup {\mathcal {U}}_{n}} , and then, for each n {\displaystyle n} and U ∈ U n {\displaystyle U\in {\mathcal {U}}_{n}} , let
C ( n , U ) = U − ∪ k < n U k . {\displaystyle C(n,U)=U-\cup _{k<n}U_{k}.}
Then C := { C ( n , U ) ∣ n , U } {\displaystyle {\mathcal {C}}:=\{C(n,U)\mid n,U\}} is a locally-closed cover; indeed, if n {\displaystyle n} is the least integer such x ∈ U n {\displaystyle x\in U_{n}} , then x {\displaystyle x} is in some U ∈ U n {\displaystyle U\in {\mathcal {U}}_{n}} and x ∉ U k , k < n {\displaystyle x\not \in U_{k},k<n} . For "locally finite", let x {\displaystyle x} be in X {\displaystyle X} ; say, x ∈ C ( n , U ) {\displaystyle x\in C(n,U)} for some n , U {\displaystyle n,U} . Since each U k {\displaystyle {\mathcal {U}}_{k}} is locally finite, there is a neighborhood V {\displaystyle V} of x {\displaystyle x} such that V {\displaystyle V} intersects only finitely many sets in ∪ k ≤ n U k {\displaystyle \cup _{k\leq n}{\mathcal {U}}_{k}} . Then U ∩ V {\displaystyle U\cap V} intersects only finitely many sets in C {\displaystyle {\mathcal {C}}} , since U {\displaystyle U} is disjoint from the sets C ( k , W ) , k > n {\displaystyle C(k,W),k>n} . ◻ {\displaystyle \square }
Proof of Proposition Let X be a T1-space in which each open cover admits a closure-preserving closed refinement. We first note that the assumption on open covers can be strengthen to the following
(*) For each open cover U i , i ∈ I {\displaystyle U_{i},i\in I} , there exists a closure-preserving closed cover F i , i ∈ I {\displaystyle F_{i},i\in I} indexed by the same I {\displaystyle I} such that F i ⊂ U i . {\displaystyle F_{i}\subset U_{i}.}
Indeed, given such an open cover U i {\displaystyle U_{i}} , by assumption, we have a closure-preserving closed refinement F {\displaystyle {\mathcal {F}}} of it. For each i ∈ I {\displaystyle i\in I} , let
G i = ⋃ { F ∈ F ∣ F ⊂ U i } . {\displaystyle G_{i}=\bigcup \{F\in {\mathcal {F}}\mid F\subset U_{i}\}.}
Then this cover G i {\displaystyle G_{i}} has required properties. Second, we shall note that X {\displaystyle X} is normal. Indeed, given disjoint closed subsets F , G {\displaystyle F,G} , the open cover { X − F , X − G } {\displaystyle \{X-F,X-G\}} has a closed refinement { X − U , X − V } {\displaystyle \{X-U,X-V\}} ; say, X − U ⊂ X − F {\displaystyle X-U\subset X-F} and then U , V {\displaystyle U,V} are disjoint neighborhoods of F , G {\displaystyle F,G} . In particular, X {\displaystyle X} is Hausdorff since X {\displaystyle X} is T1. We shall now verify the hypothesis of Lemma 1. Let U a , a ∈ A {\displaystyle U_{a},a\in A} be an open cover indexed by a well-ordered set A {\displaystyle A} . First, we shall recurviely construct a closure-preserving closed cover F a , n {\displaystyle F_{a,n}} as follows. For n = 1 {\displaystyle n=1} , let F a , 1 ⊂ U a {\displaystyle F_{a,1}\subset U_{a}} be a closed cover given by (*). Assuming we have constructed F a , n − 1 {\displaystyle F_{a,n-1}} , let
U a , n = U a − ∪ b < a F b , n − 1 . {\displaystyle U_{a,n}=U_{a}-\cup _{b<a}F_{b,n-1}.}
It is an open cover: if a {\displaystyle a} is the least element with the property x ∈ F a , n − 1 ⊂ U a {\displaystyle x\in F_{a,{n-1}}\subset U_{a}} , then x ∉ F b , n − 1 , b < a {\displaystyle x\not \in F_{b,n-1},b<a} and so x ∈ U a , n {\displaystyle x\in U_{a,n}} . By (*), let F a , n ⊂ U a , n {\displaystyle F_{a,n}\subset U_{a,n}} be a closure-preserving closed cover. Now, since X = F a , n ∪ ( ∪ b ≠ a F b . n ) {\displaystyle X=F_{a,n}\cup (\cup _{b\neq a}F_{b.n})} , we have:
V a , n := X − ( ∪ b ≠ a F b , n ) ⊂ F a , n , {\displaystyle V_{a,n}:=X-(\cup _{b\neq a}F_{b,n})\subset F_{a,n},}
which is open since { F a , n ∣ a , n } {\displaystyle \{F_{a,n}\mid a,n\}} is closure-preserving. We claim { V a , n ∣ a , n } {\displaystyle \{V_{a,n}\mid a,n\}} is an open cover. Let x ∈ X {\displaystyle x\in X} and for each n {\displaystyle n} , let a n {\displaystyle a_{n}} be the least element such that x ∈ F a n , n {\displaystyle x\in F_{a_{n},n}} . We have a 1 ≥ a 2 ≥ ⋯ {\displaystyle a_{1}\geq a_{2}\geq \cdots } since F a , n ∩ F b , n − 1 = ∅ , b < a {\displaystyle F_{a,n}\cap F_{b,n-1}=\emptyset ,b<a} . Since A {\displaystyle A} is well-ordered, the sequence a n {\displaystyle a_{n}} cannot be strictly decreasing all the way; i.e., a n − 1 = a n {\displaystyle a_{n-1}=a_{n}} for some n {\displaystyle n} and then similarly we see x ∉ F b , n , b ≠ a n . {\displaystyle x\not \in F_{b,n},b\neq a_{n}.} That is, x {\displaystyle x} is in V a n , n {\displaystyle V_{a_{n},n}} . Note also V a , n ∩ V b , n = ∅ , b ≠ a . {\displaystyle V_{a,n}\cap V_{b,n}=\emptyset ,b\neq a.}
Finally, by (*), we can find a closure-preserving closed refinement D a , n ⊂ V a , n {\displaystyle D_{a,n}\subset V_{a,n}} . By normality, we then find open sets W a , n {\displaystyle W_{a,n}} such that
D a , n ⊂ W a , n ⊂ W a , n ¯ ⊂ V a , n . {\displaystyle D_{a,n}\subset W_{a,n}\subset {\overline {W_{a,n}}}\subset V_{a,n}.}
For each n {\displaystyle n} , let U n {\displaystyle U_{n}} be the union of all open sets, each of which intersects only finitely many sets in { W a , n ∣ a } {\displaystyle \{W_{a,n}\mid a\}} . It is a neighborhood of D n := ∪ a D a , n {\displaystyle D_{n}:=\cup _{a}D_{a,n}} since { W a , n ∣ a } {\displaystyle \{W_{a,n}\mid a\}} is a disjoint collection. By normality, we have neighborhoods U n ′ {\displaystyle U'_{n}} of D n {\displaystyle D_{n}} with closures lying in U n {\displaystyle U_{n}} . Then let
W a , n ′ = W a , n ∩ U n ′ . {\displaystyle W_{a,n}'=W_{a,n}\cap U_{n}'.}
Then { W a , n ′ ∣ a } {\displaystyle \{W_{a,n}'\mid a\}} is locally finite. Thus, by Lemma 2, we find a locally finite locally-closed refinement C {\displaystyle {\mathcal {C}}} of { W a , n ′ ∣ a , n } . {\displaystyle \{W_{a,n}'\mid a,n\}.} Then, since passing to closure doesn’t spoil “locally finite", { C ¯ ∣ C ∈ C } {\displaystyle \{{\overline {C}}\mid C\in {\mathcal {C}}\}} is a refinement satisfying the hypothesis of Lemma 1. (The argument after "finally" except Lemma 2 is essentially due to Dowkey.) ◻ {\displaystyle \square }
Similar result There is a similar result due to A. H. Stone (which was actually used to prove a metric space is paracompact).
Given a set X {\displaystyle X} and a cover C {\displaystyle {\mathcal {C}}} of it, for each element x ∈ X {\displaystyle x\in X} , we write
St ( x , C ) = ⋃ { C ∈ C ∣ x ∈ C } , {\displaystyle \operatorname {St} (x,{\mathcal {C}})=\bigcup \{C\in {\mathcal {C}}\mid x\in C\},}
and call it the star over x {\displaystyle x} with respect to C {\displaystyle {\mathcal {C}}} . A collection D {\displaystyle {\mathcal {D}}} of subsets of X {\displaystyle X} is then called a barycentric refinement of C {\displaystyle {\mathcal {C}}} if { St ( x , D ) ∣ x ∈ X } {\displaystyle \{\operatorname {St} (x,{\mathcal {D}})\mid x\in X\}} is a refinement of C {\displaystyle {\mathcal {C}}} . A barycentric refinement is in particular a refinement. Lemma 1 is then strengthened to
If each open cover has a locally finite closed refinement, then each open cover has a locally finite barycentric open refinement.
Notes
References E. Michael, A note on paracompact spaces, Proc. Amer. Math. Soc. vol. 4 (1953) pp. 831-838. Michael, E. (1957), "Another note on paracompact spaces", Proceedings of the American Mathematical Society, 8 (4): 822–828, doi:10.1090/S0002-9939-1957-0087079-9, JSTOR 2033306, MR 0087079 Mathew, Akhil (August 19, 2010), "A theorem of Michael on paracompactness", Climbing Mount Bourbaki Engelking, Ryszard (1989), General Topology, Sigma Series in Pure Mathematics, vol. 6 (2nd ed.), Berlin: Heldermann Verlag, ISBN 3-88538-006-4, MR 1039321 Schubert, Horst (1968). Topology. London: Macdonald & Co. ISBN 978-0-356-02077-8. OCLC 463753. Willard, Stephen (2012), General Topology, Dover Books on Mathematics, Courier Dover Publications, ISBN 9780486131788, OCLC 829161886
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