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Countable Paracompactness and Monotonization in Topology

Explore the concepts of countable paracompactness, monotone normality, and monotonization in topology, with a focus on characterizations, operators, and implications for Moore spaces. Joint work with Lylah Haynes and Chris Good. Understand the monotonization process and how it affects metrizability in spaces.

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Countable Paracompactness and Monotonization in Topology

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  1. Monotone Countable Paracompactness Lylah Haynes HaynesL@maths.bham.ac.uk Joint work with Chris Good

  2. Countable paracompactness and d-normality A space is countably paracompact if every countable open cover has a locally finite open refinement [Dowker, Katětov]. X x [0,1] is normal iff X is countably paracompact and normal [Dowker]. A space is d-normal if every pair of disjoint closed sets, one of which is a regular Gd-set, can be separated by disjoint closed sets [Mack]. X x [0,1] is d-normal iff X is countably paracompact [Mack].

  3. Characterizing countable paracompactness 1)For every decreasing sequence (Dn)n of closed sets with empty intersection, there exists a sequence (Un)n of open sets such that DnUn for each n and [Ishikawa]. 2)For every decreasing sequence (Dn)n of closed nowhere dense sets with empty intersection, there exists a sequence (Un)n of open sets such that DnUn for each n and Ø [Hardy & Juhász]. 3)If C is a closed subset of X x [0,1] and D is a closed subset of [0,1] such that C (X x D) = Ø, then there are disjoint open sets separating C and X x D [Tamano]. 4)X x [0,1] is d-normal [Mack]. Ø

  4. Tamano’s characterization of cp 1 [ C closed in X x [0,1] D closed in [0,1] C D X x D [ 0 X

  5. What is monotonization? For example, monotone normality represented pictorially: U’ U H(C’,U’) H(C,U) C’ C

  6. MCP A space X is monotonically countably metacompact (MCM) iff there is an operator U assigning to each n and each closed set D, an open set U(n,D) such that 1)DU(n,D), 2) if DE then U(n,D) U(n,E) and 3) if (Di)i is a decreasing sequence of closed sets with empty intersection, then X is monotonically countably paracompact (MCP) if also 4) if (Di)i is a decreasing sequence of closed sets with empty intersection, then [Good, Ying] Monotone condition MCM/MCP introduced by Good, Knight and Stares Ø. Ø.

  7. Motivation for monotonization MN Set theoretic assumptions can often be abandoned, for example 1) the Normal Moore Space Conjecture is true under PMEA. 2) every monotonically normal Moore space is metrizable. MCP Set theoretic assumptions can often be abandoned, for example 1) every countably paracompact Moore space is metrizable under PMEA. 2) every MCP Moore space is metrizable.

  8. Monotonizing Tamano’s characterization A space X is MCPiff there is an operator U assigning to each pair (C,D), where C is closed in X x [0,1] and D is closed in [0,1] such that C (X x [0,1]) = Ø, an open set U(C,D) such that 1)CU(C,D) X x ([0,1] \ D) 2) if CC’ and D’D, then U(C,D) U(C’,D’). Monotone condition 1 [ U(C,D) C D X x D [ 0 X

  9. Sketch proof Suppose U is an MCP operator. Define 1 [ C r D X x D [ 0 ( ( U(nrD,Cr) Cr X

  10. Sketch proof Suppose the monotonization of Tamano’s characterization holds Then Dn U(n,(Di)) for each n, if DiEi for each i, then U(n,(Di)) U(n,(Ei)) for each n, Ø. Hence MCP. 1 D1 x {1} 1/2 D2 x {1/2} U(D,{0}) D3 x {1/3} 1/3 Dn x {1/n} 0 X x {0} ( ) U(n,(Di)) X

  11. MdN A space X is monotonically d-normal (MdN) if there is an operator H assigning to each pair of disjoint closed sets (C,D) in X, at least one of which is a regular Gd-set, an open set H(C,D) such that 1)CH(C,D) X \ D 2) if CC’ and D’D, then H(C,D) H(C’,D’). X x [0,1] MdN X MCP X x [0,1] MCP X MdN

  12. 1 [ H(C,D) C D X x D [ 0 X Proof X x [0,1] MdN X MCP Since [0,1] is metrizable, every closed set in [0,1] is a regular Gd-set. Therefore X x D is a regular Gd-set. Hence any MdN operator H satisfies the ‘monotonized Tamano’ characterization of MCP.

  13. MCP MdN F 1 point compactification of Mrówka’s Y-space F is an infinite mad family of infinite subsets of w (i.e. for every F1,F2F, F1F2 is finite, and if Sw is an infinite set not in F, there exists F’F such that SF’ is infinite). Y* is MCP since it is compact. 1) If C and D are disjoint closed sets in Y*, then one must contain at most a finite subset of F. 2) Any closed subset of Y* containing at most finitely many points of F is a regular Gd-set. Hence if Y* is MdN, it is MN and so Y is MN. Contradiction. w Y = Fw Y*= Y N( )= (Y\K), K compact in Y

  14. MdN MCP The Sorgenfrey line S S is MN, so it is both MdN and collectionwise normal. Suppose S is MCP. Then S is MCM ≡ b. Since S is a regular g-space, it is a Moore space (b + g implies developable). Since collectionwise normal Moore implies metrizable [Bing], S is metrizable. Contradiction. [ ) a b R

  15. Nowhere dense MCM and MCP A space X is nowhere dense MCM (nMCM)iff there is an operator U assigning to each n and each closed nowhere dense set D, an open set U(n,D) such that 1)DU(n,D), 2) if DE then U(n,D) U(n,E) and 3) if (Di)i is a decreasing sequence of closed nowhere dense sets with empty intersection, then Ø. X is nowhere dense MCP (nMCP) if also 4) if (Di)i is a decreasing sequence of closed nowhere dense sets with empty intersection, then Ø.

  16. nMCP vs MCP 1) nMCM ≡ MCM 2) wN ↔ nMCP + q [wN ↔ MCP + q] 3) nMCP + Moore → Metrizable [MCP + Moore → Metrizable] nMCP ≡ MCP?

  17. For more information, see the above article, Topology and its Applications 154 (2007) 734—740

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