Monotone Countable Paracompactness

Lylah HaynesHaynesL@maths.bham.ac.uk

Joint work with Chris Good

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 -normal if every pair of disjoint closed sets, one of which is a regular G-set, can be separated by disjoint

closed sets [Mack].

X x [0,1] is -normal iff X is countably paracompact [Mack].

Countable paracompactness and -normality

1) For every decreasing sequence (Dn)n of closed sets with empty intersection, there exists a sequence (Un)n of open sets such that Dn Un 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 Dn Un 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 -normal [Mack].

Characterizing countable paracompactness

nnU

nnU Ø

0

1 [

[

X

D X x D

C

Tamano’s characterization of cp

C closed in X x [0,1]

D closed in [0,1]

For example, monotone normality represented pictorially:

What is monotonization?

C

H(C,U)

U

C’

H(C’,U’)

U’

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) D U(n,D),

2) if D E 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]

MCP

Ø.

MCM/MCP introduced by Good, Knight and Stares

),(n

nDnU

n

nDnU ),( Ø.

Monotone condition

Monotone condition

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.

Motivation for monotonization

A space X is MCP iff 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) C U(C,D) X x ([0,1] \ D)

2) if C C’ and D’ D, then U(C,D) U(C’,D’).

Monotonizing Tamano’s characterization

),( DCU

0

1 [

[

X

X x D

C

D

U(C,D)

Monotone condition

Monotone condition

Suppose U is an MCP operator.

Define

Sketch proof

0

1 [

[

X x D

C

D

r

Cr

11

rDrDn

rD2

X(

(

U(nrD,Cr)

).,(),(),(]1,0[

rDrDrr

rD rrCnUDCU

Suppose the monotonization of Tamano’s characterization holds

Then Dn U(n,(Di)) for each n,

if Di Ei for each i, then U(n,(Di)) U(n,(Ei)) for each n,

Ø. Hence MCP.

Sketch proof

X

D1 x {1}

0

1

D2 x {1/2}

D3 x {1/3}

Dn x {1/n}

( )U(n,(Di))

U(D,{0})1/2

X x {0}

1/3

n

iDnU ))(,(

ii i

DD1

A space X is monotonically -normal (MN) 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 G-set, an open set H(C,D) such that

1) C H(C,D) X \ D

2) if C C’ and D’ D, then H(C,D) H(C’,D’).

MN

),( DCH

X x [0,1] MN

X MCP

X x [0,1] MCP X MN

X x [0,1] MN X MCP

Since [0,1] is metrizable, every closed set in [0,1] is a regular G-set.Therefore X x D is a regular G-set.

Hence any MN operator H satisfies the ‘monotonized Tamano’ characterization of MCP.

Proof

0

1 [

[

X

X x D

C

D

H(C,D)

MCP MN

F

1 point compactification of Mrówka’s -space

F is an infinite mad family of infinite subsets of (i.e. for every F1,F2 F, F1 F2 is finite, and if S is an infinite set not in F, there exists F’ F such that S F’ is infinite).

* is MCP since it is compact.

1) If C and D are disjoint closed sets in *, then one must contain at most a finite subset of F.

2) Any closed subset of * containing at most finitely many points of F is a regular G-set.

Hence if * is MN, it is MN and so is MN. Contradiction.

= F

*=

N( )= (\K), K compact in

}{

}{

The Sorgenfrey line S

S is MN, so it is both MN and collectionwise normal.

Suppose S is MCP. Then S is MCM ≡ . Since S is a regular -space,

it is a Moore space ( + implies developable).

Since collectionwise normal Moore implies metrizable [Bing], S is

metrizable. Contradiction.

MN MCP

[ )a b R

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) D U(n,D),

2) if D E 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.

Nowhere dense MCM and MCP

),(n

nDnU

n

nDnU ),(

1) nMCM ≡ MCM

2) wN ↔ nMCP + q [wN ↔ MCP + q]

3) nMCP + Moore → Metrizable [MCP + Moore → Metrizable]

nMCP vs MCP

nMCP ≡ MCP?

For more information, see the above article,

Topology and its Applications 154 (2007) 734—740