Download ON C O VERIN G PROPERTIES AN D GEN ERALIZED OPEN SETS

O N C O V E R IN G P R O P E R T IE S A N D G E N E R A L IZ E D O P E N
SE TS IN T O P O L O G IC A L SPA C E S
M
a x im il ia n
G
anster
(Received April 1989)
1 . In tro d u ctio n
This paper is a survey o f results concerning the properties ’compact’
and ‘Lindelof ’ in the setting of generalized open sets. Most of the results
presented have already appeared in the literature. We have, however,
included also some improvements and some new results.
Let S’ be a subset of a topological space X . We will denote the clo­
sure of S and the interior o f S with respect to X by cl.*S and in t*S
respectively. The set o f accumulation points of S is denoted by Sd. The
subset S is called nowhere dense (abbreviated nwd) if int* cl* S = 0 .
No separation axioms are assumed unless stated explicitly.
D efinition 1.1. A subset S o f a space X is called
(i)
an a-set, if S C int* ( d a i n t y S )),
(ii)
a semi-open set, if S C cl* (hit* S),
(iii) a preopen set, if S C intx(cl;rS).
These notions are due to Njastad [15], Levine [10 ] and Mashhour et al
[13] respectively. The concept o f a preopen set was introduced by Corson
and Michael [1] who used the term ‘locally dense’ . The collection of all
a-sets in a space X is a topology on the set X , called the a-topology
o f X . We will denote the set X with the a-topology by X a. In general
neither the family 5 0 ( X ) all semi-open subsets o f X , nor the family
P O (X ) o f all preopen subsets o f X is a topology on the set X . Njastad
[15] has proved that 5 0 ( X ) is a topology if and only if X is extremally
disconnected, i.e. the closure o f every open set is open. Ganster [4]
has shown that P O (X ) is a topology if and only if clx G is open and
{x } is preopen for each x € int^F where X = F U G denotes the
so-called Hewitt-representation o f X (for the definition o f the Hewittrepresentation see [4]).
Dedicated to Professor H. Florian on the occasion of his 65th birthday.
Math. Chronicle 19 (1990). 27-33
We conclude this section by stating a result on characterization of
a-sets.
P rop osition 1.2. For a subset S o f a space X the following are equiv­
alent:
(i)
S is an a-set in X ,
(ii)
S is semi-open and preopen [ 19],
(iii)
S = U — N where U is open in X and N is nwd in X [15].
2. C om pactn ess
D efinition 2.1. A space X is called a-compact (resp. semi-compact,
resp. strongly compact) if every cover o f X by a-sets (resp. semi-open
sets, resp. preopen sets) has a finite subcover.
The concept of ar-compactness was defined by Maheshwari and Thakur
[11] and further investigated by Noiri and Di Maio [16]. Dorsett [2] and
Prasad and Yadav [17] independently defined semi-compact spaces while
strongly compact spaces are defined in [14] and discussed in [8] and [5].
It is clear that every semi-compact space as well as every strongly
compace space is a-compact. The following result of Dorsett [2] charac­
terizes semi-compact spaces.
T h eorem 2.2. A space X is semi-compact if and only if every nwd
subset o f X is finite and every disjoint family o f nonempty open sets is
finite.
For further results on semi-compact spaces we refer the reader to [7].
Concerning strong compactness, the following result of Jankovic, Reilly
and Vamanamurthy [8 ] and o f Ganster [5] shows how strong this prop­
erty really is.
T h eorem 2.3. A space X is strongly compact if and only if X is com­
pact and X d is finite.
C orolla ry 2.4. A space which is strongly compact and semi-compact
has to be finite.
R em ark 2.5. Hausdorff strongly compact spaces are simply a finite
(topological) sum o f 1-point-compactifications of discrete spaces.
We now turn to a-compact spaces ([11], [12], [20 ]). By Proposition
1.2, if N is a nwd subset of a space X then N is closed and discrete in
X a. Hence we have the following result.
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T h eorem 2 .6. For a space X the following are equivalent:
(i)
(ii)
X is a-compact (countably a-compact),
X a is compact (countably compact),
(iii) X is compact (countably compact) and every nwd subset of X is
finite.
Our next result seems to be new and it shows that for Hausdorff spaces
the concepts o f a-compactness and strong compactness coincide.
T h eorem 2.7. Every Hausdorff a-compact space X is strongly com­
pact.
P ro o f. We first note that every infinite Hausdorff space contains an
infinite discrete subspace. Now consider the decomposition X = A U B
where A is closed and dense-in-itself and B is open and scattered (see
e.g. [3], page 59). Since any discrete subspace of A is nwd in X , it follows
from Theorem 2.6 that A is finite and hence empty. So X is scattered.
Then X d is nwd and thusiinite. By Theorem 2.3, X is strongly compact.
C orollary . Every Hausdorff semi-compact space is finite.
R em ark . Theorem 2.7 is false in the absence of Hausdorffness. Let X
be an infinite set with the cofmite topology. Since X = X a and since
X is compact, X is obviously a-compact. By Theorem 2.3, however, X
fails to be strongly compact. In fact, X is antistrongly compact, i.e. the
only strongly compact subspaces of X are the finite ones.
L em m a 2.10. I f X is T\ and countably a-compact then every nwd
subset of X is finite. Moreover, X = X a.
Lem m a 2.11. I f X is R0 and not compact then X contains an infinite
discrete subspace (flSj, Lemma 1).
Lem m a 2.12. I f X is countably compact and X d is countable then X
is compact.
P r o o f. It is sufficient to prove that X is Lindelof. Let {G , : i G / } be
any open cover o f X and let X d = {a „ : n G IN}. For each n G IN choose
in G / with an G Gin and let H = (J {^ u : n € IN}. Then X — H is
closed and discrete. Since X is countably compact, X — H is finite and
thus X is Lindelof.
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T h eorem 2.13. Every T\ countably a-compact space is a-compact.
P ro o f. By Lemma 2.10 we have X = X a . Consider the decomposition
X = A U B where A is closed and dense-in-itself and B is open and
scattered. Then clx B — B is nwd, hence finite and so cl\ B is scattered
and countably a-comp act. As a consequence of Lemma 2.12, c l* fl is
compact. By Lemma 2.11 A has to be compact since every discrete
subset of a dense-in-itself T\ space is nwd. Now X = A U cl x B and so
X is compact and thus a-compact since X = X a .
C orollary 2.14. Every T 2 countably a-compact space is strongly com­
pact.
3. Lindel5fness
D efinition 3.1. A space X is called a-Lindelof (resp. semi-Lindelof,
resp. strongly Lindelof) if every cover o f X by a-sets (resp. semi-open
sets, resp. preopen sets) has a countable subcover.
Clearly every semi-Lindelof space as well as every strongly Lindelof
space is a-Lindelof. The following result is easily established.
T h eorem 3.2. For a space X the following are equivalent:
(i)
X is a-Lindelof,
(ii)
X a is Lindelof,
(iii) X is Lindelof and every nwd subset o f X is at most countable.
Strongly Lindelof spaces are defined in [14] and investigated in [6].
There are, o f course, similarities between strongly Lindelof spaces and
strongly compact spaces. For example, a space X is strongly Lindelof
if and only if X is Lindelof and X d is at most countable (Theorem 2.7
in [6]). On the other hand, Example 2.10 in [6] provides an example of
a compact Hausdorff space which is strongly Lindelof but not strongly
compact.
We are now going to consider the concept of a semi-Lindelof space.
Recall that Kunen [9] defines a Luzin space to be an uncountable Haus­
dorff space having at most countably many isolated points and in which
every nwd subset is at most countable. It is very well known that the
continuum hypothesis (CH) implies that the real line has a Luzin sub­
space. We note that every Luzin space is hereditarily Lindelof (and
consequently has countable spread) [9].
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P ro p o s itio n 3.3. If X is hereditarily Lindelof and each nwd subset is
countable, then X is semi-Lindelof. In particular, every Luzin space is
semi-Lindelof.
P r o o f. Let {5 ,- : * € 1} be a semi-open cover o f X . For each * G J let
Ui be open in X such that Ui C 5< C cljct/,-. If V =
: * € / } then
V is open and dense, hence X — V is nwd and so countable. Since V is
Lindelof there exists a countable subset V of I such that X — (J{f/j :
* G I 1} U ( X — V ), hence X is covered by { X — V ] U {S* : * G / ' } . This
proves that X is semi-Lindelof.
Lem m a 3.4. Lei X be semi-Lindelof. Then X has at most countably
many isolated points.
P r o o f. Suppose there exists a subset M o f X consisting of isolated
points and card(Af) = u\. Let M = [){M p : f) < w i} with
= 0
whenever 0 ^ 7 and cwtd(Mp) = Wj for each f) < u \ . If C = cl^ M —M
then C is nwd, hence countable and we have, by setting U = X —
cljfAf, X — cljfM U U. For each x G C let f}x be the smallest /? with
x G c\xMp if such a f3 exists, otherwise let
= 0 < u>\. Let y < v i
satisfy 7 £ {0 X : x G C }. Now, if 5 = cl*([J {A f 0 : P £ 7 }) then
c lx M = S U M1 . Clearly S is semi-open and
U { { x } : x G M7 }
is a semi-open cover o f X with no countable subcover, a contradiction.
Our previous considerations now lead to the following result.
T h eorem 3.5. Lei X be an uncountable Hausdorff space. Then X is
semi-Lindelof if and only if X is a Luzin space.
C orollary. Let X be uncountable, Hausdorff and dense-in-itself. Then
the following are equivalent:
(i)
X is a-Lindelof,
(ii)
X is semi-Lindelof,
(iii)
X is Luzin space.
C orollary 3.7. Suppose that X is semi-Lindelof and strongly Lindelof.
Then X is countable.
By Theorem 3.5, uncountable Hausdorff semi-Lindelof spaces exist
under C H . On the other hand it was shown by Kunen [9] that there
are no Luzin spaces assuming Martin’s Axiom and the negation of the
continuum hypothesis. Thus we have the following.
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R em ark 3.8. The existence of uncountable Hausdorff semi-Lindelof
spaces is independent o f ZFC (i.e. Zermelo-Fraenkel set theory with the
Axiom o f Choice).
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Maximilian G anater,
Graz University of Technology,
Kopemikusgasse 24,
A-8010 Graz,
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