Series I: COMMENTATIONES MATHEMATICAE XXIX (1990) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE XXIX (1990)
Jin g c h e n g To n g
(Jacksonville)
Arens and Dugundji’s decomposition of compactness
Abstract. Arens and Dugundji proved that a topological space is compact if and only if it is countably compact and metacompact. In this paper, two generalizations of this theorem are given.
One weakens both conditions, one weakens metacompactness. We also give a decomposition of countable compactness.
1. Introduction. To bridge the gap between countable compactness and compactness, Arens and Dugundji [2] or [5] proved the following theorem:
Th e o r e m
1.1. A topological space X is compact if and only if it is countably compact and metacompact.
Aquaro [1] generalized this theorem.
Th e o r e m
1.2. A countably compact space X is compact if each open cover
‘Ш of X has a point-countable open refinement.
Generalizations of the above theorem can be found in Aull [3], Chaber [4], Wicke and Worrell [10], and Worrell and Wicke [11].
In this paper, we introduce the notions of T-space and cu-space, give a generalization of Theorem 1.1 and a generalization of Theorem 1.2, and prove a decomposition theorem for countable compactness.
2. Preliminaries. We first recall some basic facts. Let A be a subset of a topological space X. If every neighborhood of
x e Xcontains at least m points of A, then * is said to be an m-limit point of A; if every neighborhood of x contains infinitely many points of A, then x is said to be an co-limit point of A. A topological space X is said to be m-limit point compact if every infinite sequence has an m-limit point in X; a space X is said to be compact if every open cover of X has a finite subcover. Obvious
ly compact => countably compact => m-limit point compact, and for 7]-spaces, countably compact = m-limit point compact (m ^ 2) = 2-limit point compact.
A topological space X is said to be a metacompact space if every open
cover °U of X has an open refinement iT such that for each
x e X ,there are
2 1 0 J i n g c h e n g T o n g
only finitely many open sets in Y containing x. If % is an open cover of X, an open refinement °U is said to be point-countable if each x e X belongs to at most countably many open sets in Y .
Following Munkres [8], p. 254, a sequence of points {хл}*°=1 is said to be locally discrete if each point of X has a neighborhood that intersects at most one point of the sequence.
The following lemma is trivial.
Le m m a
2.1. A topological space X is 2-limit point compact if and only if it has no locally discrete sequence.
Now we introduce some notations. If °U is a collection of subsets of a topological space X, we denote the collection of all elements of % containing a point x e X by S>(x, ûïï). Then union of all elements of @{x, %) is said to be the star of x and is denoted by St(x, %). \${x, tft)\ denotes the cardinality of 9 { x , Ш).
The following two lemmas are straightforward.
Le m m a
2.2. Let °U be an open cover of a topological space X. Then for each x e X , {x} c: St(x, <Щ.
Le m m a
2.3. Let Ш be an open cover of a topological space X. I f there is an infinite sequence {x„}®=1 of X such that St(xf, %) n {x„}) = {xj, then {x„}*=1 is a locally discrete sequence.
The following lemma is a generalization of Lemma 1 in [4].
Le m m a
2.4. Let Ш be an open cover of a 2-limit point compact space X. I f Y ç X , then there are finitely many points x1? x2, ..., хие У such that
Y c z U ? = i S t ( x i 5 ^ ) .
P ro o f. If У cannot be covered by a union of stars of finitely many points of Y, we can find х 1еУ such that У ф St(xl5 ^U). Choose an x2e y \S t(x l5 we have У ф (Jf=1 St(xi5 %). Choose an x3e
y \ (J ,2=1 St(xi5 Y), and inductively we have a sequence {x„}^°=1 such that St(x,-, ^ ) n ( ( J * =l-{x„}) = {xj. By Lemma 2.3, {хи}®=1 is a locally discrete sequence, X cannot be 2-limit point compact. ■
From Lemma 2.4, we immediately obtain a generalization of Theorem 1.1.
Th e o r e m
2.1. A topological space X is compact if and only if it is 2-limit point compact and metacompact.
Lemma 2.4 also gives a generalization of Theorem 1A in [4] (cf. [7], Chapter 1, Theorem 21).
Th e o r e m
2.2. Let °U be an open cover of a 2-limit point compact space X.
I f there is a sequence {<%„}i of open covers of X such that for each x e X ,
there is an n ^ 1 with St(x, %n) a U for a certain U
e%, then °tt has a finite subcover.
Lemma 2.2 gives an interesting property of a metacompact space.
Th e o r e m
2.3. In a metacompact space X, {*} is compact for each
x eX .3. Main results.
De f in it io n
3.1. Let °U be an arbitrary open cover of a topological space X.
If there is an open refinement 'Ÿ' of 41 such that for every infinite (resp.
uncountable) closed set S cz X, there is at least one point
x eS satisfying 1 ^ \S>{x, У)\ < со (resp. 1 ^ \9{x, <%)\ < со), then X is said to be an F-space (resp. co-space). The refinement 'Ÿ' is said to be an F-refinement (resp.
co-refinement).
Obviously, a metacompact space is an F-space; a space X for which every open cover has a point-countable refinement is an co-space. Simple examples show that the converses are not true.
Now we give two lemmas.
Le m m a
3.1. Let % be an open cover of an F-space, Y be the F-refinement of
°U. I f C = {xeX ; 1 < \&>(x, У)\ < со), then either
(i) there are two finite sets I, J such that X = \J ieI St(x£, 'F') и ({Jjej Uj), where x teC, UjEtft;
or
(ii) for any finite set /, C X l J ^ S t ^ , Y') Ф 0.
P ro o f. Suppose (ii) does not hold, we have a finite set / such that С c (J ieJSt(x,-, 'V) with x teC. Then St(x£, 'f") must be a finite closed set, it can bt covered by finitely many open sets in °U. Hence (i) holds. ■
Le m m a
3.2. Let Ш be an open cover of an co-space X, 'V be the co-refinement of °U. I f C = {xeX ; 1 < \$>{x, Ÿ~)\ ^ со), then either
(i) there is a finite set I and a countable set J such that X = Uf6/ S t(*г> r ) u (LU
ju j)> where x ieC, UjE%;
or
(ii) for any finite set I, C \ \ J ieISt{xi, 'V') Ф 0.
P ro o f. Similar to the proof of Lemma 3.1 ■ Now we generalize Theorem' 1.1.
Th e o r e m
3.1. A topological space X is compact if and only if it is 2-limit point compact and is an F-space..
P roof. The necessity is trivial, we prove the sufficiency.
If X is not compact, let % be an open cover of X without finite open
refinement, and let тГ be the F-refinement of °U\ then 'V has no finite subcover.
212 J i n g c h e n g T o n g
By (ii) of Lemma 3.1, we can find an infinite sequence {x„}®=1 such that St(xi5 Y') n((j£°=i {x„}) = {xj. By Lemma 2.3, {*„}*=! is a locally discrete sequence, X cannot be 2-limit point compact, a contradiction. ■
Now we generalize Theorem 1.2.
Th e o r e m
3.2. A countably compact co-space X is compact.
P ro o f. If X is not compact, let 41 be an open cover of X without finite open refinement, and let тГ be the F-refinement of 4l\ then Y has no finite subcover. By (ii) of Lemma 3.2, we can find an infinite sequence {x„}®=i such that St (xf, Y ) n ((J “= ! {x„}) = { x j. By Lemma 2.3, {xn}*= t is a locally discrete sequence, X cannot be countably compact, a contradiction. В
The following theorem bridges the gap between 2-limit point compactness and countable compactness. The proof is a version of a theorem in [9].
Th e o r e m
3.3. A topological space X is countably compact if and only if it is 2-limit point compact and (x) is countably compact for each x e X .
P ro o f. Let 41 = {Ut , U2, Un, ...} be a countable open cover of X.
Choose an arbitrary x xeX ; then ( x j can be covered by a finite subset 4lx of 41. Let nl be the maximal index i of U ^41^. If X cannot be covered by a finite subcover of 41, then there is an x2e 2 f \( J " i1 Ut, and (x2) can be covered by a finite subset of 412 of 41. Let n2 be the maximal index i of l/£e (J£=i 411. Then if X cannot be covered by a finite subcover of 41, we can find an x 3e X \{ J ? L l l/f. Thus inductively we can find a sequence {x„}®=1 such that { x j can be covered by a finite subset 41 „ of 41 and xk+ x t Uti where nk is the maximal index i of U (J*=14lt.
Now we prove that {x„}®=1 is a locally discrete sequence.
Let x e X be an arbitrary point. Then x e U p for a certain Upe 41. It is easily seen that there is an nq such that nq- 1 < p ^ n q. Hence Up n ([jrL p+l (x,)) = 0. We discuss two possible cases: (i) хф\J ? =1 (xf) . Then Up\(J ? =1 {xf} is an open neighborhood of x containing no points of the sequence (x„}®=1; (ii) x e [J ? =1 ( x j. Then x e { x j for a certain t < q. Without loss of generality we may assume that t is the minimal index of the set {t; x e{x t}}. Hence x ^ { x j for i < t — 1. For i ^ t+ l and i ^ q, notice that Ш c U?= i u i and xt
eX \ U P
i u j^ X \ U;= i 4
j l. we have_W c
X \ U f - i Uj since ^ \U ? = i Uj *s a cl°sed set. Hence {xj n ((Jf= (+1 (x j) = 0.
Therefore, x e { x j, x £ (J? =f+1 { x j, UP\[ J j$ qj * p {Xj} is an open neighbour
hood of x containing only one point xt of the sequence {x„}®=1. Therefore {x„}£°=i is a locally discrete sequence, X cannot be 2-limit point compact. This contradiction proves the sufficiency. В
Theorem 3.3, together with Theorem 2.3, gives a good explanation of
Theorem 2.1.
Acknowledgement. The author would like to thank the referee for his very valuable suggestions to improve this paper.
References
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DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY O F NORTH FLORIDA
JACKSONVILLE, FL 32216, U.S.A.
7 — Roczniki PTM — Prace Matematyczne XXIX
