Minimal Wallman covers of Tychonoff spaces

J.Korean Math. Soc. 34 (1997), No. 4, pp. 1009-1018

MINIMAL WALLMAN COVERS OF TYCHONOFF SPACES

CHANG IL KIM

ABSTRACT. Observing that for any{3c-Wallman functorA and any Tychonoff space X, there is a cover (CI(A(X), X), Cl) of X such that X is A-disconnectedifand only ifCl : CI(A(X), X) ^---t X is a homeomorphism, we show that every Tychonoff spacehas the minimal A-disconnected cover. We also show that if X is a weakly LindelOf or locally compact zero-dimensional space, then the mini- mal G-disconnected (equivalently, cloz)-cover is given by the space CI(A(X), X) which is a dense subspace of Eec ({3X).

1. Introduction

All spaces in this paper are Tychonoff spaces and f3X denotes the Stone-Cech compactification of a space X.

It is known that minimal covers of some spaces are given by certain filter spaces. Indeed, the minimal extremally disconnected cover EX of a space X is given by the filter space {a : a is a fixed R(X)-ultrafilter}

([5]) and for any locally weakly Lindelof space X, the minimal basically disconnected (quasi-F, resp.) cover of X

is

characterized by the filter space AX = {a : a is a fixed aZ(X)#-ultrafilter} (QF(X) = ^{{a: a is}

a fixed Z(X)#-ultrafilter}, resp.) ([6]). In [4], it is shown that every comapct space X has a minimal cloz-cover (Ecc(X),

W'G(X))

and in [8]

([3], resp.), a theory of basically disconnected (quasi-F, resp.) covers of Tychonoff spaces is developed and the relation between AX and Af3X (QF(X) and QF (f3X), resp.) is explored. Henriksen, Vermeer, and Woods ([4]) introduced the notion of Wallman sublattices and showed that for any compact space X and Wallman sublattice A(X)

Received July 8, 1997.

1991 Mathematics Subject Classification: MClO, 54D80, 54G05.

Key words and phrases: covering map, A-disconnected space, minimal cover.

This work was supported by GRANT No. KOSEF 951-0100-001-2 from the Korea Science and Engineering Foundation.

of R(X), X has a Wallman cover £(A(X), X) of A(X). In fact, for any compact space X, AX.

=

£(o-Z(X)#), QF(X)

=

£(Z(X)#), and Ecc(X)

=

£(G(X), X) (see [8], [3], and [4]).

Let Tychc be the category of Tychonoff spaces and covering maps and Lat the category of lattices and lattice homomorphisms. In this paper, we first introduce the notion of tJc-Wallman functor A: Tychc

----t

Lat, that is, A is a contravariant functor such that for any X and f

: Y

^----t

X in Tychc, A(X) is a Wallman sublattice ofR(X), A(,6X)x =

{A n X : A E A(,6X)} = A(X), A(J)(A) = cly(inty(J-l(A») (A

E

A(X» , and a space X is A-disconnected (that is, for any A, B

^E

A(X) with A

1\

B = 0, A n B = 0) jf and only if tJX is A-disconnected. We show that for any ,6c-Wallman functor A, every space has the minimal A-disconnected cover and if X is a weakly Lindelof or locally com- pact zero-dimensional space, then the minimal G-disconnected (equiv- alently, cloz) cover is given by the space C1(G(X), X) which is a dense subspace of E cc(,6X). For the terminology, we refer to [1] and [2].

2. tJ-Wallman sublattices

Recall that the collection R(X) of all regular closed sets in a space X, when partially ordered by inclusion, becomes a complete Boolean algebra, in which the join, meet, and complementation operations are defined as follows: if A

^E

R(X) and

{~

:

i E

I}

~

R(X), then

V{Ai :

ⁱ

El}

⁼

clx(U{Ai :

ⁱ

El}),

I\{Ai :

ⁱ

El}

=

clx(intx(n{Ai:

i E

I}», and A'

=

clx(X - A)

and a sublattice of R(X) is a subset of R(X) that contains 0, X and is closed under finite joins and meets.

DEFINITION

2.1. ([4]) Let X be a space and A(X) a sublattice of R(X). Then

(a) A(X) is said to be T1 with respect to X if for any A ^E A(X) and x

~

A, there is B

E

A(X) such that x

E

intx(B) and A

1\

B

=

0,

(b) A(X) is said to be normal if for any A, B

E

A(X) with A

1\

B

=

0, there are C, D in A(X) such that A

1\

C

=

B

1\

D

=

0 and C U

D

=

X, and

Minimal Wallman covers of Tychonoff spaces 1011

(c) if A(X) satisfies (a) and (b) , then A(X) is said to be a Wallman sublattice of R(X).

NOTATION 2.2. For any space X, let

(a) Z(X)# = {clx(intx(A» : A is a zero-set in X}, and

(b) O"Z(X)# denote the smallest O"-complete Boolean subalgebra of R(X) containing Z(X)#, and

(c) G(X) = {clx(C) : C is a cozero-set set in X and there is a cozero-set D

in

X such that CnD = 0 and CUD is dense in X}.

It is well-known that for any space X, R(X), Z(X)#, O"Z(X)#, and G(X) are Wallman sublattices of R(X) ([4]).

DEFINITION 2.3. Let (L, ::;) be a lattice with the top element 1 and the bottom element

0

and F

_~

L. Then F is said to be

(a) an

L-filter

if (1) 0 # F.

(2) for any a,

b

E F, there is c E F with 0 # ^{c ::; a}

^1\b.

(3) if a E F and a ::; bEL, then bE F.

(b) an

L-ultrafilter

if F is a maximal L-filter.

Let X be a compact space, A(X) a Wallman sublattice of R(X) , and £(A(X» = {a : a is an A(X)-ultrafilter on X}. For any A E A(X), let A* = {a E £(A(X» : A E a}. Then {A* : A E A(X)} is a base for closed sets of some compact topology on £(A(X». And let

£(A(X), X) be the subspace {(a, x) E £(A(X»

x

X : x E na} of the product space £(A(X»

x

X. Then the map

^{WA(X) :}

£(A(X), X)

^---7

X

(WA(X)(a,

x) = x) is a covering map, that is, a continuous perfect irreducible map, and A(X) is a base for closed sets

in

X if and only if the map

k :

£(A(X), X)

^{- - 7}

£(A(X» defined by

k«a,

x» = a

is

a homeomorphism ([4]).

Recall that for any dense subspace X of a space Y, the map

</J :

R(Y)

---7

R(X) defined by

</J(

A)

=

A n X (A E R(Y» is a Boolean isomorphism.

For any space X and A(X)

~

R(X), let A(X)

(3

denote the set {A E R(.BX) : A n ^X

^E

A(X)}.

DEFINITION 2.4. Let X be a space and A(X) a Wallman sublattice

of R(X). Then A(X) is said to be ,B-Wallman if A(X)(3 is a Wallman

sublattice of R(,BX).

Let X be a space, A(X) a (3-Wallman sublattice of R(X), and Cl (A(X»

=

{o:: : 0:: is a fixed A(X)-ultrafilter on X}. For any A

E

A(X), let A+

=

{o::

E

Cl(A(X» : A

E

o::}. Then {A+ : A

E

A(X)} is a base for closed sets of some topology on Cl(A(X). And let Cl(A(X), X) be the subspace {(0::, x)

E

Cl (A(X» x X : x

E

no::} of the product space Cl (A(X» x X and W A(X);3x : W.:AtX);3 (X)

^{- - - t}

X the restriction and corestriction of the map W A(X);3 : £(A(X)j3, (3X)

^{- - - t}

(3X with respect to W:AtX);3 (X) and X, respectively. Then W A(X),I3x is a covering map.

PROPOSITION 2.5. Let X be a space and A(X) a (3- Wallman sub- lattice ofR(X). Then there

is

a homeomorphism h : Cl(A(X), X)

^{- - - t}

W:AtX);3 (X) and hence the map

Cl =

W A(X);3x

0

h from

Cl

(A(X), X) to X is a covering map.

Proof

Since A(X) and A(X)j3 are lattice isomorhpic, for any A(X)- ultrafilter 0::, there

is

a unique A(X)j3-ultrafilter 0::13

=

{A

E

A(X)j3 : A n X

E

o::} and vice-versa. Let T =W:AtX);3(X), Define a map h : Cl(A(X), X)

^{- - - t}

T by h((o::,x»

=

(0::13, x). Then h

is

one-to-one and onto.

Let A

E

A(X) and U be an open set in (3X. Then there

is

a unique Aj3

E

A(X)j3 with A

=

Aj3 n X. Let G

=

[(Cl(A(X» - A+) x (U n X)]

n Cl(A(X),X) and H

=

(£(A(X),B) -

A~)

x U) n T. Take any (r, y)

^E

G. Then'Y rt. A+; 'Y,B rt.

A~.

So h((r, y»

=

(r,B, y)

E

(£(A(X),B) -

A~)

x U. Since WA(X);3((r,B,y» = y

E

X, h(b,y» = b,B,y)

E

T. Hence h(G)

~

H.

Let (813' z)

E

H. Since z

E

X and z

E

n8, h- l ((8j3, z»

=

(8, z)

E

G.

Hence h(G)

=

H. Thus h is a homeomorphism and Cl

=

WA(X);3x

⁰

h

is a covering map. 0

DEFINITION 2.6. Let Tych

c

be the category of Tychonoff spaces and covering maps and

Lat

the category of lattices and lattice homo- morphisms. Then a contravariant functor A : Tych

c ^{- - - t} Lat

is said to be (3-Wallman if

(a) for any X

E

Tych

_{c ,}

A(X) is a (3-Wallman sublattice of R(X) and A(X) 13 = A({3X) ,

(b) for any f : Y

^{- - - t}

X in Tych

_{c ,}

A(J)(A)

=

cly(inty(J-l(A»)

(A

E

A(X».

Minimal Wallman covers of Tychonoff spaces 1013

Let R (Z#, aZ#, G, resp.) : Tych

_c ^{- - - t}

Lat be a contravariant functor taking each Tychonoff space X to the lattice R(X) (Z(X)#, O"Z(X)#, G(X), resp.) and each covering map f :

^Y ~

X to the lattice homomorphism A(J) : A(X)

^{- - - t}

A(Y) for which A(J)( A)

=

cly(inty(J-l(A») (A

E

A(X»), where A = R, Z#, O"Z#, or G, resp.

Then R, Z#, O"Z#, and G are 13- Wallman functors.

DEFINITION

2.7. Let A be a 13- Wallman functor and X a space.

Then X is called A-disconnected if for any A, B

E

A(X) with A

1\

B

=

0, An B

=

0.

Recall that a space X is called extremally disconnected (basically

disconnected, cloz,

resp.) if every element in R(X) (O"Z(X)#, G(X), resp.) is clopen in X and that a space X is called quasi-F if for any A,

B E

Z(X)#, A

1\B=

A n

B.

Hence a space X is extremally disconnected (basically disconnected, quasi-F, doz, resp.) space if and only if it is R (O"Z#, Z#, G, resp.)-disconnected.

THEOREM

2.8. Let A be a 13- Wa11man functor and X a space. Then X is A-disconnected if and only

ifCl : Cl

(A(X), X)

~

X is a home- omorphism.

Proof

(=}) It is enough to show that Cl is one-to-one. Take any (a,x)

-=J

(-y,y) in C1(A(X), ^{X). If} x

-=J y,

then cl((a,x»

=

x

-=J y =

Cl ((" y». Suppose that a

-=J "

then there are A, B

E

A(X) such that A

E

a,

BE,

and A

1\

B

=

0. Since X is A-disconnected, A n

B =

0

and hence cl((a,x»

=

x

=1=Y=

Cl((T,y». So Cl is one-to-one.

(

~)

Suppose that there are A, B

E

A(X) such that A

1\

B

=

0 and An

B -=J

0. Pick x

^E

An B. Let aD = {E

E

A(X) : x

^E

intx(E)}U{A}.

Then there is an A(X)-ultrafilter a ^with aD

~

a. Suppose that x

~

na, then x

~

E for some E E a. Since A(X) is T

1

with respect to X, there is F

E

A(X) with x

E

intx (F) and E

1\

F

=

0. Since F

E

a, we have a contradiction. Hence x

E

na. Similarly there is a ,

E Cl

(A(X» with x

E

n, and

BE,.

Hence (a, x)

=1=

(-y, x) and so

Cl

is not one-to-one.

Thus we have a contradiction. 0

3. Minimal A-disconnected covers

Note that a space X

is

R (O"Z#, Z#, G, resp.)-disconnected if and

only if fJX

is

also R (aZ#, Z#, G, resp.)-disconnected

DEFINITION 3.1. A fJ-Wallman funetor A is said to be fJc-Wallman if for any A-disconnected space X, fJX is A-disconnected.

Recall that a pair (Y, J) is called a cover of a space X if f : Y

----+

X is a covering map.

DEFINITION 3.2. Let A be a fJ-Wallman functor and X a space.

Then

(a) a cover (Y, J) of X is called an A-disconnected cover of X if Y is A-disconnected,

(b) an A-disconnected cover (Y, f) of X is called a minimal A- disconnected cover of X if for any A-disconnected cover (K, g) of X, there is a covering map h : K

^---T

Y with foh

= g.

LEMMA 3.3. ([4]) Let X be a compact space and A(X) a Wallman sublattice of R(X). Then (£ (A(X), X), WACX))

is

a cover of X such that

(a) for any A, B E A(X) with A /\ B =0, c1.cCAcx),X)(WA~X)(intx(A)))

n c1.cCAcx),X)(WA~X)(intx(B)))

⁼

0, and

(b) if f : Y

----+

X is a covering map such that for any A, B

E

A(X) with A /\ 8

=

0, c1.cCACX),X) (f-l(intx(A))) n c1.cCACX) ,X) (f-l (intx (B)))

= 0, then there

is

a covering map

g:

Y

----+

£(A(X), X) with f

⁼

WACX)

0 g.

For any fJc-Wallman functor A and space X, the following diagram

£( A(fJX) , fJX )

x

13 1

\[JA(;3X) ---'---'-7)

fJX.

is a pullback in Top, where

j

is a dense embedding and Top is the category of topological spaces and continuous maps. Using this and the above lemma, we have the following:

THEOREM 3.4. Let A be a fJc- Wallman functor and X a space. If

(Y, f)

is

an A-disconnected cover of X, then there

is

a covering map 9

: Y

^----+

Cl(A(X), X) with

Clog=

f.

Minimal Wallman covers of Tychonoff spaces 1015 COROLLARY

3.5. Let A be a 13c- Wallman functor and X a space. If C

1

(A(X), X) is A-disconnected, then (C

1

(A(X), X), Cl) is the minimal A-disconnected cover of X.

In the following, for any space X, let (EX, Jrx) denote the minimal R-msconnected cover (absolute, or minimal extremally disconnected cover) of X. Indeed, for any space X, (C

1

(R(X), X),

Cl)

is the minimal R-disconnected cover of X ([5]).

Note that for any 13- Wallman functor A and X

E

Tych

_{c ,}

(EX, Jrx) is an A-disconnected cover of X.

THEOREM

3.6. Let A be a 13c- Wallman functor. Then for any space X, there is the mininmal A-disconnected cover (C{A(X), X),

^CA(X»)

of X.

Proof Let Co(A(X), X) = ^{X, cg} = Ix and a be an ordinal. Suppose that for any ordinal 13 ^{with j3} < a,

(i) for any ordinal, with, ::; 13, there is a cover (C"Y(A(X) , X), cri) of X, and

(ii) for any ordinals d" with d < , ::; 13, there is a covering map cJ : C,,(A(X), X)

^---+

C8(A(X), X) with cri = cg

⁰

cJ.

For any ordinals ,1, ,2, ,3 ^{with'l <} ,2 ^< ,3 ::; ^{13, since cri}

^{3 =}

^cril

⁰

C"Y3 "Y1 = C"Yl

0 ⁰

(C"Y1 "Y2

⁰

C"Y3) "Y2 and C"1

0

is a covering map '''Y1 C"Y3

=

C"Y2

^{")'1 0}

C"Y3.

^")'2

Let a be a non-limit ordinal, then there is an ordinal 13 with a = 13 + 1. Let Ca(A(X), X)

=

C

1

(A(CJ3(A(X), X)), CJ3(A(X), X)) and

c~

=

Cl : Ca(A(X), X)

---+

CJ3 (A(X), X). Then (i) and (H) hold for a.

Let a be a limit ordinaL Let I

=

{13 : 13 ^is an ordinal with j3 < a}.

Define an inverse limit system D : I

---+ Top

as follows: for any j3 and

, < j3 in I, let D(j3) = CJ3(A(X), X) and Db ::; (3) = S : C}3(A(X) , X)

---+

C"Y(A(X), X). Let (Ca(A(X), X),

c~)J3<a

be the inverse limit of D. Then (i) and (H) hold for a.

By Theorem 3.4 and the transfinite induction, for any ordinal a, there is a covering map fa : EX

---+

Ca(A(X), X)) with Co

⁰

fa =

Jr

x. Hence all spaces Ca (A(X), X) lie between X and EX and therefore there is the smallest ordinal ao such that

C~~+l ^:

Cao+l(A(X), X)

---+

Cao(A(X), X) is a homeomorphism. By Theorem 2.8, Cao(A(X), X) is an A-disconnected space. Let C(A(X), X) = Cao(A(X), X) and

CA(X)

= ego : C(A(X), X)

---+

X. By Theorem 3.4 and the transfinite

induction, (C(A(X), X), CA(X)) is the mininmal A-disconnected cover

~X

0

Let A and S be fJc-functors such that for any space X, SeX)

<;;;;

A(X).

If a space X is A-disconnected, then X is also S-disconnected. Using this and the fact that if f,g, and h are covering maps with fog

=

foh, then 9

=

h, we have the following:

PROPOSITION

3.7. Let A and S be /3c-functors such that for any space X, SeX)

<;;;;

A(X). Then for any space X, there is a homeomor- phism h: C(A(Y), Y)

~

C(A(X), X) with CB(X)OCA(y)

=

CA(x)oh, where Y

=

C(S(X), X).

Recall that a space X is weakly Lindelof if every open cover U of X has a countable subfamily V of U with clx(UV) = X and that a space X is called locally weakly Lindelof if every element of X has a weakly Lindelof neighborhood in X.

In [6), it is shown that for any locally weakly Lindelof space X, (Cl(Z(X)#, X), Cl) ((Cl((TZ(X)#, X), Cl), resp.) is the minimal Z#

((TZ#, resp. )-disconnected cover of X. Now we will try to find suitable conditions for a space X for which (Cl(G(X), X), Cl) is the minimal G-disconnected cover of X.

LEMMA

3.8. Let A be a /3- Wallman functor such that for any open subspace S of a space Y, A(Y)s

=

{A n S : A

E

A(Y)}

<;;;;

A(S). Then a space X is A-disconnected if and only if every element of X has an A-disconnected open neighborhood in X.

Proof. Suppose that every element of X has an A-disconnected open neighborhood in X. Let A, B

^E

A(X) with A /\ B

=

0. Suppose that there is an x

E

A n B. Let S be an A-disconnected open neighborhood of x in X. Note that A n S, B n S

E

A(S) and (A n S) /\ (B n S) =0.

Since S is A-disconnected, A n B n S

=

0. This is a contradiction.

Hence X is an A-disconnected space. The converse is trivial. 0

Recall that a space X is a G-disconnected (equivalently, cloz-) space

if and only if every dense cozero-set C in X is D-extendable in X, that

is, for any continuous map f : C

--+

D, there is a continuous map

9 :

X

--+

D with gle

⁼

f, where D denotes the two point discrete space

([4]) and that for any dense weakly Lindelof subspace X of a space Y

and A

E

Z(X)#, there is B

E

Z(Y)# with A = B n X.

Minimal Wallman covers of Tychonoff spaces 1017 LEMMA

3.9. If X is a elopen or dense weakly Linde18f subspace of a G-disconnected space Y, then X is also G-disconnected

Proof. Let X be a subspace of a G-disconnected space Y. Suppose that X is clopen in Y and take any dense cozero-set C in X. Then C is a cozero-set in

Y

and hence C

U (Y -

X) is a dense cozero-set in Y. Since C and Y - X are disjoint c10pen sets in C

U

(Y - X), C is D-extendable in C

U

(Y - X). Sinec Y is G-disconnected, C

U

(Y - X) is D-extendable in

Y

and hence C is D-extendable in Y. Thus X is G-disconnected.

Suppose that X is a dense weakly LindelOf subspace of Y. Let D

₁ E

G(X), then there is a cozero-set

D2

in X such that D

₁

n

D 2 =

0 and D

1 U D₂

is dense in X. Since X is a dense weakly Lindelof subspace of Y, there are cozero-sets E

b

E

2

in Y such that D

i ~

cly(E

i )

(i

=

1,2), El n E

2 =

0 and El

U

E

2

is dense in

Y.

Hence c1y(E

1 )

n c1y (E

_{2 )} =

0 and thus c1y (D

1 )

n c1y(D

2 )

= 0. Therefore X is G-disconnected. 0

In [4], the minimal G-disconnected cover of a compact space X is characterized by (L:(G(X), X), wGCX»). In the literature, for a compact space X, L:(G(X), X) is denoted by Ecc(X).

It

is known that if f : Y

^---t

X is a covering map and X is a weakly Lindelof space, then Y is weakly Lindelof ([3]).

THEOREM

3.10. IfX is a locally compact zero-dimensional or weakly Linde18f space, then (C

₁

(G(X), X),

Cl)

is the niinimal G-disconnected cover ofX.

Proof. Let Z = wGC!3X) : EccCBX)

^----7

f3X and W = Z-l(X) .

Suppose that X is a locally compact zero-dimensional space. Take

any yEW, then there is a c10pen compact neighborhood B of i(y) in

X. Since B is c10pen in /3X, Z-l(B) is c10pen in E cc (/3X) and Z-l(B)

^~

W. By Lemma 3.9, Z-l(B) is a G-disconnected open neighborhood of

y

in W. Thus, by Lemma 3.8, W is G-disconnected and therefore, by

Corollary 3.5, (Cl (G(X), X), Cl) is the minimal G-disconnected cover of

X.

If

X is a weakly Lindelof space, then W is a weakly Lindelof subspace

of Ecc (/3X) and hence by Corollary 3.5 and Lemma 3.9, (Cl (G(X), X),

Cl) is the minimal G-disconnected cover of X. 0

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J.

Vermeer and R. G. Woods, Quasi-F-covers of Tychonoff spaces, 'frans. Amer. Math. Soc.303 (1987), 779-804.

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Department of Mathematics Education Dankook University

Seoul 140-714, Korea