Z-CLOSED SPACES

Travis Thompson

(received 23 March, 1981; revised 30 April, 1981)

A completely regular Hausdorff space (Tychonoff) is basically dis­connected if and only if the closure of every cozero set is open [2].In this paper we investigate basically disconnected spaces using a type of convergence and obtain characterisations of compact basically dis­connected spaces. All spaces in this paper are assumed to be Tychonoff.

Definition 1. A filterbase F = : aeA) z-convergee to a pointoxqgX if for each zero set Z such that x0cZ , there exists an

oanth such that A c Z .

<*0

Definition 2. A filterbase F = z-accumulatee to a point xqcXo oif for each zero set Z such that xqeZ and every ^ e F , A^ D Z t $ .

Note that in the definitions above Z need not be a zero neigh­bourhood set of ; i.e., ac0 may not be an interior point of Z .

Definition 3. A topological space X is Z-closed if and only if foro

every collection {Z } of zero sets such that X = U Z , there exists 1 a ao

a finite subcollection such that X = U Za . t

Definition 4. A topological space X is S-cloeed if and only if for every semiopen cover of X there exists a finite subfamilysuch that the union of their closures is X [3].

Math. Chronicle 11(1982) Part 2 103-107.

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Evidently, every S-closed space is Z-closed since Z^ is a semi­open set for each zero set Z^ .

Theorem 1. Let F be a maximal filterbase in X . Then F z-accumu- latea to a point x0eX if and only if F z-converges to Xq .

Theorem 2. For a topological space X t the following are equivalent:

(i) X is Z-closed;

(ii) For each family of cozero sets {U } such thato

fl - <t> , there exists a finite subfamilyo

{U } such that fl V = 4> ; a . a .v i

(iii) Every filterbase z-accumulates to some point x0tX ;

(iv) Each maximal filterbase z-converges.

Theorem 3. If Y is a cozero subset of a Z-closed space X , then o_¥ is a Z-closed subset of X .

Theorem 4. A basically disconnected compact space X is Z-closed.

The proofs of Theorems 1-4 are routine and are omitted.

A Tychonoff space X is a P-space if and only if each zero set is open. A space X is an F-space if and only if every cozero set is C*-embedded. A space X is a quasi-F space if and only if each dense cozero set is C*-embedded. An extremally disconnected space is a space in which every open subspace is C*-embedded. An almost-P-space is a space in which every zero set has non-empty interior. In general, we have the following diagram of implications:

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extremally disconnected

IP-space basically disconnected ---- ► F-space

I JAlmost P-space Quasi F-space

Example 1. Let U be a free ultrafilter on the positive integers N , and let Z = N U {0} , a/.N . Equip Z with the following topology: all points of N are isolated, and the open sets containing o are the sets U U (a) , where i/eli . Then Z is extremally disconnected, hence 6(E) is S-closed [7»] and 6(Z) is Z-closed, where 6(Z) is the Stone-Cech compactification of Z •

Example 2. Let X be any P-space. Then B(^) is basically dis­connected and compact, hence a Z-closed space ( X is basically dis­connected if and only if B(A') is basically disconnected).

Example 3. Let S be an uncountable set and ezS . Define a topology on S as follows: All points, except e , are isolated; a neighbour­hood of 8 is any set A c 5 such that seA and S-A is countable. Then S is a P-space but is not extremally disconnected. Therefore, 8(S) is Z-closed but neither an S-closed space nor a P-space (compact P-spaces are finite).

Example 4. Let T be the topological sum of Z and s . Then T is basically disconnected, but neither extremally disconnected nor a P-space, but B(T) is Z-closed.

Theorem 5. A first countable Z-closed space is finite.

Proof. A Z-closed space is a quasi-F space. By [l, Prop. 5.5], a first countable quasi-F space is discrete, hence a discrete Z-closed space must be finite.

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Theorem 6. If X is a normal Z-closed space, then X is basically disconnected.

_ SLProof. Suppose there exists a cozero set U such that U - V f <t>

_ _ oand X - U t <f> . Let xeU - U . Then for every open set VeAf(x) ,V fl U f <t> . Therefore, F = {K fl U | KeiV(x)} forms a filterbase.Since X is Z-closed, F has a 2-accumulation point X qcX . Also,F converges to x in the usual sense. Suppose, if possible, that

o oxqe.* - U . Then setting Z = X U , we have that x q e X - U = Z .

75Since F 3-accumulates to x0 , (V fl U) fl Z ̂<f> , an impossibility,

o oTherefore, x q £X - U and X qeU . Since X is completely regular, '

_ _ othere exists cozero sets W and G such that x 0eW < ^ W ^ G ^ G ^ U . Since W 0 (X-G) = 4> , there exist zero set neighbourhoods Z(x) and Z(x0) such that W c Z(xq) , (.X-G) c Z(x) , and

o oZ(x0) fl Z(x) = <f> . Evidently, XqcZ(xq) and xcZ(x) . Therefore, since F converges to x , there exists (F(x) fl U) e F such that

o o(P(x) fl U) c Z(x) ; but (K(x) fl U) fl (Z(x0)) = <t> , contradictingthe fact that F 2-accumulates to Xq • Therefore, our supposition

_ o _is false and either U - U = <j> or X - U = <t> . In either case, theclosure of U is open and X is basically disconnected.

Corollary. Let X be a compact space. X is Z-closed if and only if X is basically disconnected.

Corollary. For a normal topological space X % the following are equivalent:

(i) X is basically disconnected;

(ii) The lattice C(X) is conditionally a-complete (i.e., every countable family with upper bound in C(X)

has a supremum in C{X)) ;

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(iii) The lattice C(8(AT)) is conditionally a-complete;

(iv) B(X) is basically disconnected;

(v) B(*) is Z-closed.

The equivalence of (i) through (iv) is well known.

This last useful corollary allows us to see that 0(/V) - N is not Z-closed since it is not basically disconnected. But B(/V) - N is an almost P-space. Hence, a compact almost P-space is not necessarily Z-closed, and a compact quasi-F space need not be Z-closed.

Questions. Can the requirement of normality be dropped from Theorem 6? When are Z-closed spaces productive?

REFERENCES

1. F. Dashiell, A. Hager, and M. Henriksen, Order-Cauchy Completions of Rings and Vector Lattices of Continuous Functions, Can. J. Math., 32(1980), 657-685.

2. L. Giliman and M. Jerison, Rings of Continuous Functions.D. Van Nostrand Co., Inc., Princeton, N.J., 1960.

3. T. Thompson, S-closed Spaces, Proc. of Amer. Math. Soc., 60(1976), 335-338.

Louisiana State University

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