Almost Menger property in bitopological spaces
We introduce the notion of almost Menger property in bitopological spaces. We give some characterizations in terms of $(i, j)$-regular open sets and almost continuous surjection. We also investigate the notion of almost $\gamma$ -set in the bitopological context.
Saved in:
| Date: | 2016 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2016
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1882 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507765250719744 |
|---|---|
| author | Eysen, A. F. Özçağ, E. Ейсен, А. Е. Озчаг, Е. |
| author_facet | Eysen, A. F. Özçağ, E. Ейсен, А. Е. Озчаг, Е. |
| author_sort | Eysen, A. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:30:37Z |
| description | We introduce the notion of almost Menger property in bitopological spaces. We give some characterizations in terms of $(i, j)$-regular open sets and almost continuous surjection. We also investigate the notion of almost $\gamma$ -set in the bitopological context. |
| first_indexed | 2026-03-24T02:14:31Z |
| format | Article |
| fulltext |
UDC 517.9
S. Özçağ, A. E. Eysen (Hacettepe Univ., Ankara, Turkey)
ALMOST MENGER PROPERTY IN BITOPOLOGICAL SPACES
МАЙЖЕ МЕНГЕРОВА ВЛАСТИВIСТЬ У БIТОПОЛОГIЧНИХ ПРОСТОРАХ
We introduce the notion of almost Menger property in bitopological spaces. We give some characterizations in terms of
(i, j)-regular open sets and almost continuous surjection. We also investigate the notion of almost \gamma -set in the bitopological
context.
Введено поняття майже менгерової властивостi в бiтопологiчних просторах, наведено деякi характеристики в
термiнах (i, j)-регулярних вiдкритих множин i майже неперервної сюр’єкцiї та вивчено поняття майже \gamma -множини
в бiтопологiчному контекстi.
1. Introduction. In this paper we will be mostly concerned with the notion of almost Mengerness
in bitopological context.
We first recall the classical definition of Menger property in topological spaces: A topological
space (X, \tau ) is Menger if for each sequence \langle \scrU n : n \in \BbbN \rangle of open covers of X, there exists a
sequence \langle \scrV n : n \in \BbbN \rangle such that for every n \in \BbbN , \scrV n is a finite subset of \scrU n and
\bigcup
n\in \BbbN \scrV n is a
cover of X [20].
Hurewicz [8] introduced the Menger property in 1925 and showed that a conjecture of Menger
is equivalent to the statement that a metrizable space has the Menger property if and only if it is
\sigma -compact. In 1927 Hurewicz [9] introduced a stronger version of the Menger property called the
Hurewicz property defined as follows. A space X has the Hurewicz property if for each sequence \langle \scrU n :
n \in \BbbN \rangle of open covers of X there exists a sequence \langle \scrV n : n \in \BbbN \rangle such that for every n \in \BbbN , \scrV n
is a finite subset of \scrU n and each element of the space belongs to all but finitely many of the sets\bigcup
n\in \BbbN \scrV n. It is known that \sigma -compactness implies the Hurewicz property in all finite powers and that
the Hurewicz property implies the Menger property.
Recently many papers related to the Menger property appeared in the literature. In [22] it is
proved that there exists a non-meager, non-CDH (countable dense homogeneous) filter. In [26]
Scheepers showed that if X is a separable metric space with the Hurewicz covering property, then
the Banach – Mazur game played on X is determined and the implication is not true when Hurewicz
covering property is replaced by Menger covering property. In [2] T. Banakh and D. Repovs defined
and studied universal nowhere dense and universal meager sets in Menger manifolds.
In 1996 M. Scheepers gave general definition of selection principles and began a systematic study
of selection principles in topology. For selected results see [16, 24, 25, 29].
Many topological properties are defined or characterized in terms of the following two classical
selection principles.
Let \scrA and \scrB be sets consisting of families of subsets of an infinite set X. Then:
\sansS 1(\scrA ,\scrB ) is the selection hypothesis: for each sequence \langle An : n \in \BbbN \rangle of elements of \scrA there is
a sequence \langle bn : n \in \BbbN \rangle such that for each n, bn \in An, and \{ bn : n \in \BbbN \} is an element of \scrB .
\sansS fin(\scrA ,\scrB ) denotes the selection hypothesis: for each sequence \langle An : n \in \BbbN \rangle of elements of \scrA
there is a sequence \langle Bn : n \in \BbbN \rangle of finite sets such that for each n, Bn \subset An, and
\bigcup
n\in \BbbN Bn \in \scrB .
c\bigcirc S. ÖZÇAĞ, A. E. EYSEN, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6 835
836 S. ÖZÇAĞ, A. E. EYSEN
When both \scrA and \scrB are the collection \scrO of open covers of a space X then \sansS 1(\scrO ,\scrO ) denotes
the Rothberger property [23], while \sansS fin(\scrO ,\scrO ) is the Menger property [8, 10, 20].
Recently several papers on weaker forms of the Menger property have been published (see
[1, 12, 21]).
Kočinac in [14, 15] introduced the notion of almost Menger property and Kocev in [12] has
studied systematically this notion by giving characterizations in terms of regular open sets and almost
continuous mappings. On the other hand, weakly Menger property was introduced in [3] and studied
widely in [1, 21].
Both of these properties are weaker than the Menger property, and obviously every Menger space
is almost Menger and every almost Menger space is weakly Menger. In [13] Kocev found conditions
under which the properties Menger, almost Menger and weakly Menger are equivalent. On the other
hand, there are a very few papers dealing with these concepts in bitopological context.
However, the systematic study began in [17] by studying selective versions of separability in
bitopological spaces and continued in [18] by investigating selection principles in bitopological
spaces and proposing possible lines of investigation in this direction.
In [18] the authors defined three versions of the Menger property in a bitopological space
(X, \tau 1, \tau 2), which are \delta 2-Menger, (1, 2)-almost Menger and (1, 2)-weakly Menger. In this pa-
per we will focus on the almost Menger property in bitopolological spaces and leave the weakly
Menger property for another work.
Recall that \scrO , \Omega , \Gamma denote the families of open covers, \omega -covers and \gamma -covers of a space,
respectively.
An open cover U of X is an \omega -cover, if for each finite subset F of X there exists U \in U such
that F \subseteq U and X is not a member of U.
A topological space (X, \tau ) is a \gamma -set if for each sequence \langle \scrU n : n \in \BbbN \rangle of \omega -covers of X there
exists a sequence \langle Vn : n \in \BbbN \rangle such that for every n \in \BbbN , Vn \in \scrU n and \scrU = \{ Vn : n \in \BbbN \} is a \gamma -
cover of X (i.e., \scrU is an open cover of X such that it is infinite and for every x \in X the set \{ U \in U :
x /\in U\} is finite).
2. Definitions and examples. For undefined topological notions we refer to [6], while for
undefined bitopological notions we refer to [5].
Throughout the paper (X, \tau 1, \tau 2) will be a bitopological space, i.e., the set X endowed with two
topologies \tau 1 and \tau 2. For a subset A of \mathrm{I}\mathrm{n}\mathrm{t}\tau i(A) and \mathrm{C}\mathrm{l}\tau i(A) will denote the interior and the closure
of A in (X, \tau i) (i = 1, 2) respectively. By \tau i-open covers of X we mean that the cover of X by
\tau i-open sets in X.
In [14], Kočinac introduced the following definition.
Definition 2.1 [14]. A topological space (X, \tau ) is almost Menger, if for each sequence \langle \scrU n :
n \in \BbbN \rangle of open covers of X, there exists a sequence \langle \scrV n : n \in \BbbN \rangle such that for every n \in \BbbN , \scrV n is
a finite subset of \scrU n and
\bigcup
\{ \scrV \prime
n : n \in \BbbN \} is a cover of X, where \scrV \prime
n = \{ \mathrm{C}\mathrm{l} (V ) : V \in \scrV n\} .
The following definition was first introduced in [18].
Definition 2.2. A bitopological space (X, \tau 1, \tau 2) is said to be (i, j)-almost Menger (i, j = 1, 2,
i \not = j) if for each sequence \langle \scrU n : n \in \BbbN \rangle of \tau i-open covers of X, there exists a sequence \langle \scrV n :
n \in \BbbN \rangle of finite families such that for each n, \scrV n \subseteq \scrU n and X =
\bigcup
n\in \BbbN (
\bigcup
V \in \scrV n
\mathrm{C}\mathrm{l}\tau j (V )).
We note that if (X, \tau 1) is almost Menger and \tau 2 \leq \tau 1, then the bitopological space (X, \tau 1, \tau 2) is
(1, 2)-almost Menger.
Proposition 2.1. Let (X, \tau 1, \tau 2) be a bitopological space. If (X, \tau 1) is Menger, then (X, \tau 1, \tau 2)
is (1, 2)-almost Menger.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
ALMOST MENGER PROPERTY IN BITOPOLOGICAL SPACES 837
Proof. Obvious.
Example 2.1. Let the real line \BbbR be endowed with the Euclidean topology \tau 1 and the Sorgenfrey
topology \tau 2. Since (\BbbR , \tau 1) is Menger, the bitopological space (\BbbR , \tau 1, \tau 2) is (1, 2)-almost Menger by
Proposition 2.1. On the other hand, (\BbbR , \tau 2) is not Menger.
In the following example we show that the converse of Proposition 2.1 is not true in general.
Example 2.2. Let X be the Euclidean plane, \tau 1 is the Sorgenfrey topology and \tau 2 is the usual
topology on X. It is easy to prove that the bitopological space (X, \tau 1, \tau 2) is (1, 2)-almost Menger,
but (X, \tau 1) does not have the Menger property because X is not Lindelöff.
We can introduce the (i, j)-almost Rothberger property in a similar way.
Definition 2.3. A bitopological space (X, \tau 1, \tau 2) is said to be (i, j)-almost Rothberger (i, j =
= 1, 2, i \not = j) if for each sequence \langle \scrU n : n \in \BbbN \rangle of \tau i-open covers of X, there is a sequence \langle Un :
n \in \BbbN \rangle such that for each n, Un \in \scrU n and X =
\bigcup
n\in \BbbN \mathrm{C}\mathrm{l}\tau j (Un).
It is obvious that every (i, j)-almost Rothberger bitopological space is (i, j)-almost Menger.
In the following example we will see that the reverse implication is not true in general.
Example 2.3. There is a (i, j)-almost Menger bitopological space which is not (i, j)-almost
Rothberger.
Let the real line \BbbR be endowed with two topologies: one is the usual metric topology \tau 1, and the
other one is open-minus countable topology \tau 2 (U \in \tau 2 if and only if U = G \setminus C, where G is open
in the usual metric topology \tau 1 on \BbbR and C is a countable subset of \BbbR ) (see [28]).
The bitopological space (\BbbR , \tau 1, \tau 2) is (1, 2)-almost Menger. It follows from the fact that (\BbbR , \tau 1)
is Menger and by Proposition 2.1.
To show that (\BbbR , \tau 1, \tau 2) is not (1, 2)-almost Rothberger. Let \langle \scrU n : n \in \BbbN \rangle be a sequence of
\tau 1-open covers of \BbbR such that for each n \in \BbbN and U \in \scrU n \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}d(U) < 1/2n.
Recall that the usual metric d(x, y) = | x - y| and \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}d(U) = \mathrm{s}\mathrm{u}\mathrm{p}\{ | x - y| | x, y \in U\} . Now
we have \mathrm{C}\mathrm{l}\tau 1(U) = \mathrm{C}\mathrm{l}\tau 2(U) for each n \in \BbbN and U \in \scrU n. If for every n \in \BbbN we choose an element
Un of \scrU n, then \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}d (\cup n\in \BbbN \mathrm{C}\mathrm{l}\tau 2(Un)) \leq
\sum \infty
n=1
1/2n = 1. This establishes that (\BbbR , \tau 1, \tau 2) is not
(1, 2)-almost Rothberger.
3. Some properties. As we mentioned in the previous section, if (X, \tau 1) is Menger, then the
bitopological space (X, \tau 1, \tau 2) is (1, 2)-almost Menger. However the converse of this statement is
not true.
Now it is natural to ask in which class of spaces the reverse implication holds.
First let us recall the following definition of (i, j)-regular bitopological space and its characterizations.
Definition 3.1 [27]. A bitopological space (X, \tau 1, \tau 2) is said to be (i, j)-regular (i, j = 1, 2,
i \not = j) if for each point x \in X and each \tau i-closed set F with x /\in F, there exist \tau i-open set U and
\tau j -open set V such that x \in U, F \subseteq V and U \cap V = \varnothing .
We may state at once:
Proposition 3.1 [27]. A bitopological space (X, \tau 1, \tau 2) is (i, j)-regular if and only if for each
point x \in X and each \tau i-open set U with x \in U, there exists \tau i-open set V such that x \in V \subseteq
\subseteq \mathrm{C}\mathrm{l}\tau j (V ) \subseteq U.
Now we have the following theorem.
Theorem 3.1. If (X, \tau 1, \tau 2) is (i, j)-almost Menger and (i, j)-regular bitopological space, then
(X, \tau i) is Menger.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
838 S. ÖZÇAĞ, A. E. EYSEN
Proof. We consider only the case i = 1, j = 2. Let \langle \scrU n : n \in \BbbN \rangle be a sequence of \tau 1-open
covers of X. Since (X, \tau 1, \tau 2) is (1, 2)-regular bitopological space, for each n \in \BbbN there exists a
\tau 1-open cover \scrU \prime
n such that \mathrm{C}\mathrm{l}\tau 2(\scrU \prime
n) = \{ \mathrm{C}\mathrm{l}\tau 2(U \prime ) : U \prime \in \scrU \prime
n\} is a refinement of \scrU n.
Since (X, \tau 1, \tau 2) is (1, 2)-almost Menger bitopological space, there exists a sequence \langle \scrV \prime
n :
n \in \BbbN \rangle such that for each n, \scrV \prime
n is a finite subset of \scrU \prime
n and
\bigcup
n\in \BbbN \mathrm{C}\mathrm{l}\tau 2(\scrV \prime
n) is a cover of X,
where \mathrm{C}\mathrm{l}\tau 2(\scrV \prime
n) = \{ \mathrm{C}\mathrm{l}\tau 2(V \prime ) : V \prime \in \scrV \prime
n\} . For every n \in \BbbN and V \prime \in \scrV \prime
n we can choose UV \prime \in \scrU n
such that \mathrm{C}\mathrm{l}\tau 2(V
\prime ) \subseteq UV \prime , since \mathrm{C}\mathrm{l}\tau 2(V
\prime ) \in \mathrm{C}\mathrm{l}\tau 2(\scrU \prime
n) and \mathrm{C}\mathrm{l}\tau 2(\scrU \prime
n) refines \scrU n. Let \scrV n = \{ UV \prime :
V \prime \in \scrV \prime
n\} . Then \langle \scrV n : n \in \BbbN \rangle is a sequence with \scrV n a finite subset of \scrU n for each n \in \BbbN and\bigcup
n\in \BbbN \scrV n is a \tau 1-open covers of X.
Now instead of using \tau i-open (\tau j -open) sets we will use (i, j)-regular open sets to characterize
the (i, j)-almost Menger property in bitopological spaces.
First, let us recall the definition of (i, j)-regular open ((i, j)-regular closed) sets.
Definition 3.2 [11, 27]. Let (X, \tau 1, \tau 2) be a bitopological space. A set A \subseteq X is called (i, j)-
regular open ((i, j)-regular closed) (i \not = j, i, j = 1, 2) if A = \mathrm{I}\mathrm{n}\mathrm{t}\tau i \mathrm{C}\mathrm{l}\tau j (A) (A = \mathrm{C}\mathrm{l}\tau i \mathrm{I}\mathrm{n}\mathrm{t}\tau j (A)).
A is said to be pairwise regular open (pairwise regular closed) if it is both (i, j)-regular open and
(j, i)-regular open ((i, j)-regular closed and (j, i)-regular closed).
Clearly every (i, j)-regular open set in (X, \tau 1, \tau 2) is \tau i-open.
Theorem 3.2. A bitopological space (X, \tau 1, \tau 2) is (i, j)-almost Menger if and only if for each
sequence \langle \scrU n : n \in \BbbN \rangle of covers of X by (i, j)-regular open sets, there exists a sequence \langle \scrV n :
n \in \BbbN \rangle of finite families such that for each n \in \BbbN , \scrV n \subseteq \scrU n and X =
\bigcup
n\in \BbbN (
\bigcup
V \in \scrV n
\mathrm{C}\mathrm{l}\tau j (V )).
Proof. We consider only the case i = 1, j = 2.
(\Rightarrow ) Since every (1, 2)-regular open set in (X, \tau 1, \tau 2) is \tau 1-open, it is obvious.
(\Leftarrow ) Let \langle \scrU n : n \in \BbbN \rangle be a sequence of \tau 1-open covers of X. For each n, let \scrU \prime
n = \{ \mathrm{I}\mathrm{n}\mathrm{t}\tau 1 \mathrm{C}\mathrm{l}\tau 2(U) :
U \in \scrU n\} . Then \langle \scrU \prime
n : n \in \BbbN \rangle is a sequence of covers of X by (1, 2)-regular open sets.
By the hypothesis there exists a sequence \langle \scrV \prime
n : n \in \BbbN \rangle such that for every n \in \BbbN , \scrV \prime
n is a finite
subset of \scrU \prime
n and
\bigcup
n\in \BbbN \mathrm{C}\mathrm{l}\tau 2(\scrV \prime
n) is a cover of X, where \mathrm{C}\mathrm{l}\tau 2(\scrV \prime
n) = \{ \mathrm{C}\mathrm{l}\tau 2(V \prime ) : V \prime \in \scrV \prime
n\} . For
each n \in \BbbN and V \prime \in \scrV \prime
n there exists UV \prime \in \scrU n such that V \prime = \mathrm{I}\mathrm{n}\mathrm{t}\tau 1 \mathrm{C}\mathrm{l}\tau 2(UV \prime ). Since \mathrm{C}\mathrm{l}\tau 2(UV \prime )
is a (2, 1)-regular closed set we have \mathrm{C}\mathrm{l}\tau 2(V
\prime ) = \mathrm{C}\mathrm{l}\tau 2 \mathrm{I}\mathrm{n}\mathrm{t}\tau 1 \mathrm{C}\mathrm{l}\tau 2(UV \prime ) = \mathrm{C}\mathrm{l}\tau 2(UV \prime ). Thus for each
n \in \BbbN there is a finite subset \scrV n = \{ UV \prime : V \prime \in \scrV \prime
n\} of \scrU n such that X =
\bigcup
n\in \BbbN
\bigcup
V \in \scrV n
\mathrm{C}\mathrm{l}\tau 2(V ).
In [21] it was shown that almost Menger spaces are preserved by almost continuous mappings.
Let us now investigate the preservation of (i, j)-almost Mengerness under the almost continuous
surjective functions.
We therefore begin by recalling the concept of (i, j)-almost continuous functions.
Definition 3.3 [19]. A function f : (X, \tau 1, \tau 2) \rightarrow (Y, \sigma 1, \sigma 2) is said to be (i, j)-almost continu-
ous if f - 1(B) is \tau i-open set in X for every (i, j)-regular open set B in Y. In addition, f is called
p-almost continuous, if it is (1, 2)- and (2, 1)-almost continuous.
However we give the following theorem.
Theorem 3.3. Let (X, \tau 1, \tau 2) be (i, j)-almost Menger bitopological space and (Y, \sigma 1, \sigma 2) be a
bitopological space. If f : X \rightarrow Y is p-almost continuous surjection, then (Y, \sigma 1, \sigma 2) is (i, j)-almost
Menger.
Proof. We consider only the case i = 1, j = 2. Let \langle \scrU n : n \in \BbbN \rangle be a sequence of covers
of Y by (1, 2)-regular open sets. Let \langle \scrU \prime
n : n \in \BbbN \rangle be a sequence defined by \scrU \prime
n = \{ f - 1(U) :
U \in \scrU n\} . Since f is (1, 2)-almost continuous each \scrU \prime
n is a cover of X by \tau 1-open sets. Apply the
fact that X is (1, 2)-almost Menger; there is a sequence \langle \scrV \prime
n : n \in \BbbN \rangle of finite families such that for
each n \in \BbbN , \scrV \prime
n \subseteq \scrU \prime
n and X =
\bigcup
n\in \BbbN
\bigcup
V \prime \in \scrV \prime
n
\mathrm{C}\mathrm{l}\tau 2(V
\prime ).
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
ALMOST MENGER PROPERTY IN BITOPOLOGICAL SPACES 839
For each V \prime \in \scrV \prime
n we can choose UV \prime \in \scrU n such that V \prime = f - 1(UV \prime ). Let \scrV n = \{ UV \prime :
V \prime \in \scrV \prime
n\} . Now we have a sequence \langle \scrV n : n \in \BbbN \rangle of finite families such that for each n \in \BbbN ,
\scrV n \subseteq \scrU n and Y =
\bigcup
n\in \BbbN
\bigcup
\{ \mathrm{C}\mathrm{l}\sigma 2(UV \prime ) : V \prime \in \scrV \prime
n\} .
Now let us prove that
\bigcup
n\in \BbbN \mathrm{C}\mathrm{l}\sigma 2(\scrV n), where \mathrm{C}\mathrm{l}\sigma 2(\scrV n) = \{ \mathrm{C}\mathrm{l}\sigma 2(UV \prime ) : V \prime \in \scrV \prime
n\} is a cover of Y.
Indeed, if y = f(x) \in Y, then there exists n \in \BbbN and UV \prime \in \scrV n such that x \in \mathrm{C}\mathrm{l}\tau 2(f
- 1(UV \prime )). Since
UV \prime \subseteq Y is (1, 2)-regular open we have \mathrm{C}\mathrm{l}\sigma 2(UV \prime ) = \mathrm{C}\mathrm{l}\sigma 2 \mathrm{I}\mathrm{n}\mathrm{t}\sigma 1 \mathrm{C}\mathrm{l}\sigma 2(UV \prime ). Since Y \setminus \mathrm{C}\mathrm{l}\sigma 2(UV \prime )
is (2, 1)-regular open, f is (2, 1)-almost continuous f - 1(Y \setminus \mathrm{C}\mathrm{l}\sigma 2(UV \prime )) = X \setminus f - 1(\mathrm{C}\mathrm{l}\sigma 2(UV \prime )) is
\tau 2-open and f - 1(\mathrm{C}\mathrm{l}\sigma 2(UV \prime )) is \tau 2-closed. Then \mathrm{C}\mathrm{l}\tau 2(f
- 1(UV \prime )) \subseteq f - 1(\mathrm{C}\mathrm{l}\sigma 2(UV \prime )). This means
y \in \mathrm{C}\mathrm{l}\sigma 2(UV \prime ). Hence (Y, \sigma 1, \sigma 2) is (1, 2)-almost Menger.
For k \in \BbbN , the power bitopological space Xk of a bitopological space (X, \tau 1, \tau 2) is defined as
(Xk, \tau 1
k, \tau 2
k) in [4].
Theorem 3.4. Let (X, \tau 1, \tau 2) be a bitopological space. For each n \in \BbbN , (Xn, \tau 1
n, \tau 2
n) is
(1, 2)-almost Menger if and only if for each sequence \langle \scrU n : n \in \BbbN \rangle of \tau 1-\omega -covers of X there exists
a sequence \langle \scrV n : n \in \BbbN \rangle such that for every n \in \BbbN , \scrV n is a finite subset of \scrU n and for every finite
set F \subseteq X, there exists n \in \BbbN and V \in \scrV n such that F \subseteq \mathrm{C}\mathrm{l}\tau 2(V ).
Proof. (=\Rightarrow ) Let \langle \scrU n : n \in \BbbN \rangle be a sequence of \tau 1-\omega -covers of X. Let \{ Nm : m \in \BbbN \}
be a partition of \BbbN such that for each m \in \BbbN , Nm is infinite. For each m \in \BbbN and each
i \in Nm, let \scrU i
m = \{ Um : U \in \scrU i\} . Then \langle \scrU i
m : i \in Nm\rangle is a sequence of \tau 1
m-open covers of
Xm. Since (Xm, \tau 1
m, \tau 2
m) is (1, 2)-almost Menger there exists a sequence \langle \scrV i
m : i \in Nm\rangle such
that for every i \in Nm, \scrV i
m is a finite subset of \scrU i
m and
\bigcup
i\in Nm
\bigcup
\mathrm{C}\mathrm{l}\tau m2 (\scrV i
m) = Xm where
\mathrm{C}\mathrm{l}\tau m2 (\scrV i
m) = \{ \mathrm{C}\mathrm{l}\tau m2 (V ) : V \in \scrV i
m\} .
On the other hand, for every V \in \scrV i
m, m \in \BbbN , i \in Nm, pick an element UV \in \scrU i such that
V = UV
m. Then \langle \scrV n : n \in \BbbN \rangle is the desired sequence, where \scrV i = \{ UV : V \in \scrV i
m\} is a finite
subfamily of \scrU i.
Indeed, let F = \{ x1, . . . , xk\} be a finite subset of X. Since
\bigcup
i\in Nk
\mathrm{C}\mathrm{l}\tau k2
(\scrV i
k) is a cover of Xk
there exists an i \in Nk and V \in \scrV i
k such that (x1, . . . , xk) \in \mathrm{C}\mathrm{l}\tau k2
(V ). On the other side, we have
\mathrm{C}\mathrm{l}\tau k2
(V ) = \mathrm{C}\mathrm{l}\tau k2
(UV
k) = (\mathrm{C}\mathrm{l}\tau 2(UV ))
k for some UV \in \scrV i. Thus F \subseteq \mathrm{C}\mathrm{l}\tau 2(UV ) for UV \in \scrV i.
(\Leftarrow =) Let k \in \BbbN be fixed and \langle \scrU n : n \in \BbbN \rangle be a sequence of \tau k1 -open covers of Xk, where each
\scrU n = \{ Unm : m \in In\} .
Let F be a finite subset of X. Since F k is a finite subset of Xk, for each n \in \BbbN , there exists a
finite subset IFn \subseteq In such that F k \subseteq
\bigcup
m\in IFn Unm. In this case, there exists a \tau 1-open set VF such
that F \subseteq VF and V k
F \subseteq
\bigcup
m\in IFn Unm.
Thus \scrV n = \{ VF : F \subseteq X is finite\} is a \tau 1-\omega -open cover of X. By assumption there exists a
sequence \langle \scrW n : n \in \BbbN \rangle such that for each n \in \BbbN , \scrW n is a finite subset of \scrV n and each finite set
T \subseteq X there exists n \in \BbbN and W \in \scrW n such that T \subseteq \mathrm{C}\mathrm{l}\tau 2(W ).
Let for each n \in \BbbN , \scrW n = \{ VFj : j \in Jn\} , where Jn is a finite index set and let \scrK n = \{ Unm :
m \in Hn\} , where Hn = \{ m \in In : m \in I
Fj
n , j \in Jn\} . Then we have \scrK n is a finite subset of \scrU n and\bigcup
n\in \BbbN \mathrm{C}\mathrm{l}\tau k2
(\scrK n) is a cover of Xk, where \mathrm{C}\mathrm{l}\tau k2
(\scrK n) = \{ \mathrm{C}\mathrm{l}\tau k2 (Unm) : m \in Hn\} .
To see,
\bigcup
n\in \BbbN \mathrm{C}\mathrm{l}\tau k2
(\scrK n) is indeed a cover of Xk, now let x = (x1, . . . , xk) \in Xk. Then
T = \{ x1, . . . , xk\} is a finite subset of X. So there exists n \in \BbbN and W \in \scrW n such that T \subseteq \mathrm{C}\mathrm{l}\tau 2(W ).
Let W = VFj for some j \in Jn. Then we have
T k \subseteq (\mathrm{C}\mathrm{l}\tau 2(VFj ))
k \subseteq \mathrm{C}\mathrm{l}\tau k2
(V k
Fj
) \subseteq
\bigcup
m\in I
Fj
n
\mathrm{C}\mathrm{l}\tau k2
(Unm).
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
840 S. ÖZÇAĞ, A. E. EYSEN
So, there exists m \in I
Fj
n \subseteq Hn such that x \in \mathrm{C}\mathrm{l}\tau k2
(Unm). Thus (Xk, \tau 1
k, \tau 2
k) is (1, 2)-almost
Menger.
4. Almost \bfitgamma -set in bitopological spaces.
Definition 4.1. Let (X, \tau 1, \tau 2) be a bitopological space and \scrU be an infinite \tau i-open cover of
X. If for every x \in X the set \{ U \in \scrU : x /\in \mathrm{C}\mathrm{l}\tau j (U)\} is finite, then we call \scrU is (i, j)-almost
\gamma -cover for X.
Gerlits and Nagy [7] defined the notion of \gamma -set and Kocev in [12] defined the notion of almost
\gamma -set and its characterizations. Now let us give the definition of almost \gamma -set for bitopological spaces.
Definition 4.2. A bitopological space (X, \tau 1, \tau 2) is said to be (i, j)-almost \gamma -set if for each
sequence \langle \scrU n : n \in \BbbN \rangle of \tau i-\omega -covers of X, there exists a sequence \langle Vn : n \in \BbbN \rangle such that for all
n \in \BbbN , Vn \in \scrU n and the set \{ Vn : n \in \BbbN \} is an (i, j)-almost \gamma -cover for X.
Theorem 4.1. (X, \tau 1, \tau 2) is (i, j)-almost \gamma -set if and only if for each sequence \langle \scrU n : n \in \BbbN \rangle of
\tau i-\omega -covers of X by (i, j)-regular open sets, there exists a sequence \langle Vn : n \in \BbbN \rangle such that for all
n \in \BbbN , Vn \in \scrU n and the set \{ Vn : n \in \BbbN \} is an (i, j)-almost \gamma -cover for X.
Proof. We consider only the case i = 1, j = 2.
(=\Rightarrow ) Since every (1, 2)-regular open set in (X, \tau 1, \tau 2) is \tau 1-open, it is obvious.
(\Leftarrow =) Let \langle \scrU n : n \in \BbbN \rangle be a sequence of \tau 1-\omega -covers of X. So the set \scrU \prime
n = \{ \mathrm{I}\mathrm{n}\mathrm{t}\tau 1 \mathrm{C}\mathrm{l}\tau 2(U) :
U \in \scrU n\} is a \tau 1-\omega -cover of X, by (1, 2)-regular open sets. By the hypothesis there exists a sequence
\langle V \prime
n : n \in \BbbN \rangle such that for each n \in \BbbN , V \prime
n \in \scrU \prime
n and \scrV \prime = \{ V \prime
n : n \in \BbbN \} is a (1, 2)-almost \gamma -cover
of X.
For each n \in \BbbN , there exists Vn \in \scrU n such that V \prime
n = \mathrm{I}\mathrm{n}\mathrm{t}\tau 1 \mathrm{C}\mathrm{l}\tau 2(Vn). It is easy to check that the
sequence \scrV = \{ Vn : n \in \BbbN \} witnesses for (X, \tau 1, \tau 2) is (1, 2)-almost \gamma -set.
Theorem 4.2. Let (X, \tau 1, \tau 2) be an (i, j)-almost \gamma -set and (Y, \sigma 1, \sigma 2) be a bitopological space.
If f : X \rightarrow Y is an (i, j)-almost continuous surjection, then (Y, \sigma 1, \sigma 2) is an (i, j)-almost \gamma -set.
Proof. We consider only the case i = 1, j = 2. Let \langle \scrU n : n \in \BbbN \rangle be a sequence of \sigma 1-\omega -covers of
Y by (1, 2)-regular open set. Since f is (1, 2)-almost continuous surjection, the set \scrU \prime
n = \{ f - 1(U) :
U \in \scrU n\} is a \tau 1-\omega -cover of X.
Since (X, \tau 1, \tau 2) is (1, 2)-almost \gamma -set there exists a sequence \langle U \prime
n : n \in \BbbN \rangle such that for
each n \in \BbbN , U \prime
n \in \scrU \prime
n and \scrU \prime = \{ U \prime
n : n \in \BbbN \} is an (1, 2)-almost \gamma -cover for X. On the other
hand, for all n \in \BbbN there exists Un \in \scrU n such that U \prime
n = f - 1(Un). We claim that \scrU = \{ Un :
n \in \BbbN \} is an (1, 2)-almost \gamma -cover for Y.
Since \scrU \prime is infinite, so is \scrU . If y \in Y there exists x \in X such that f(x) = y and \{ U \prime
n \in \scrU \prime :
x /\in \mathrm{C}\mathrm{l}\tau 2(U
\prime
n)\} is finite since \scrU \prime is (1, 2)-almost \gamma -cover for X. That means there exists M \in \BbbN
such that \forall n \in \BbbN n > M, we have x \in \mathrm{C}\mathrm{l}\tau 2(U
\prime
n). For all n > M, y = f(x) \in f(\mathrm{C}\mathrm{l}\tau 2(U
\prime
n)). Since
\mathrm{C}\mathrm{l}\tau 2(U
\prime
n) \subseteq f - 1(\mathrm{C}\mathrm{l}\sigma 2(Un)) we have for each n > M, y \in \mathrm{C}\mathrm{l}\sigma 2(Un) so \{ Un \in \scrU : y /\in \mathrm{C}\mathrm{l}\sigma 2(Un)\}
is finite, hence (Y, \sigma 1, \sigma 2) is (1, 2)-almost \gamma -set.
5. Concluding remarks. The following definition of (i, j)-weakly Menger bitopological space
was first introduced in [18].
Definition 5.1. A bitopological space (X, \tau 1, \tau 2) is said to be (i, j)-weakly Menger, i, j = 1, 2,
i \not = j, if for each sequence \langle \scrU n : n \in \BbbN \rangle of \tau i-open covers of X, there exists a sequence \langle \scrV n :
n \in \BbbN \rangle of finite families such that for each n, \scrV n \subseteq \scrU n and X = \mathrm{C}\mathrm{l}\tau j (
\bigcup
n\in \BbbN \scrV n).
It would be interesting to investigate the properties of (i, j)-weakly Mengerness in bitopologi-
cal context and the relations between (i, j)-almost Menger and (i, j)-weakly Menger bitopological
spaces.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
ALMOST MENGER PROPERTY IN BITOPOLOGICAL SPACES 841
Acknowledgements. The authors would like to thank Ljubiša D. R. Kočinac for introducing this
problem and also for several useful remarks and suggestions which led to an improvement of the paper.
References
1. Babinkostova L., Pansera B. A., Scheepers M. Weak covering properties and infinite games // Topology and Appl. –
2012. – 159, № 17. – P. 3644 – 3657.
2. Banakh T., Repovs D. Universal nowhere dense and meager sets in Menger manifolds // Topology and Appl. – 2014. –
161. – P. 127 – 140.
3. Daniels P. Pixley – Roy spaces over subsets of the reals // Topology and Appl. – 1988. – 29, № 1. – P. 93 – 106.
4. Datta M. C. Projective bitopological spaces // J. Austral. Math. Soc. – 1972. – 13. – P. 327 – 334.
5. Dvalishvili B. P. Bitopological spaces: theory, relations with generalized algebraic structures and applications //
North-Holland Math. Stud. – 2005.
6. Engelking R. General topology. – Berlin: Heldermann-Verlag, 1989.
7. Gerlits J., Nagy Zs. Some properties of C(X), I // Topology and Appl. – 1982. – 14. – P. 151 – 161.
8. Hurewicz W. Über die Verallgemeinerung des Borelschen Theorems // Math. Z. – 1925. – 24. – S. 401 – 425.
9. Hurewicz W. Über Folgen Stetiger Funktionen // Fund. Math. – 1927. – 9. – P. 193 – 204.
10. Just W., Miller A. W., Scheepers M., Szeptycki P. J. The combinatorics of open covers (II) // Topology and Appl. –
1996. – 73. – P. 241 – 266.
11. Khedr F. H., Alshibani A. M. On pairwise super continuous mappings in bitopological spaces // Int. J. Math. and
Math. Sci. – 1991. – 14, № 4. – P. 715 – 722.
12. Kocev D. Almost Menger and related spaces // Mat. Vesn. – 2009. – 61. – P. 173 – 180.
13. Kocev D. Menger-type covering properties of topological spaces // Filomat. – 2015. – 29, № 1. – P. 99 – 106.
14. Kočinac Lj. D. R. Star – Menger and related spaces. II // Filomat. – 1999. – 13. – P. 129 – 140.
15. Kočinac Lj. D. R. Star – Menger and related spaces // Publ. Math. Debrecen. – 1999. – 55. – P. 421 – 431.
16. Kočinac Lj. D. R. Selected results on selection principles // Proc. Third Sem. Geometry Topology, Tabriz, Iran, 15 – 17
July, 2004. – P. 71 – 104.
17. Kočinac Lj. D. R., Özçağ S. Versions of separability in bitopological spaces // Topology and Appl. – 2011. – 158. –
P. 1471 – 1477.
18. Kočinac Lj. D. R., Özçağ S. Bitopological spaces and selection principles // Proc. Int. Conf. Topology and Appl.
(ICTA 2011). – 2012. – P. 243 – 255.
19. Maheshwari S. N., Prasad R. Semi open sets and semi continuous function in bitopological spaces // Math. Notes. –
1977/78. – 26. – P. 29 – 37.
20. Menger K. Einige Überdeckungssätze der Punktmengenlehre // Sitzungsber. Österr. Adad. Wiss. Math.-naturwiss. Kl.
Abt IIa. – 1924. – 133. – P. 421 – 444.
21. Pansera B. A. Weaker forms of the Menger property // Quaest. Math. – 2012. – 35. – P. 161 – 169.
22. Repovs D., Zdomskyy L., Zhang S. Countable dense homogeneous filters and the Menger covering property // Fundam.
Math. – 2014. – 224, № 3. – P. 233 – 240.
23. Rothberger F. Eine Verscharfung der Eigenschaft C // Fund. Math. – 1938. – 30. – P. 50 – 55.
24. Scheepers M. Combinatorics of open covers (I): Ramsey theory // Topology and Appl. – 1996. – 69. – P. 195 – 202.
25. Scheepers M. Selection principles and covering properties in topology // Note Mat. – 2003/2004. – 22, № 2. –
P. 3 – 41.
26. Scheepers M. Selection principles and baire spaces // Mat. Vesn. – 2009. – 61, № 3. – P. 195 – 202.
27. Singal A. R., Arya S. P. On pairwise almost regular spaces // Glasnik Math. – 1971. – 26, № 6. – P. 335 – 343.
28. Steen L. A., Seebach J. A. Counterexamples in topology. – New York: Dover Publ., 1995.
29. Tsaban B. Some new directions in infinite-combinatorial topology // Set Theory / Eds J. Bagaria and S. Todorčević:
Trends Math. – 2006. – P. 225 – 255.
Received 16.02.14,
after revision — 31.10.15
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
|
| id | umjimathkievua-article-1882 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:14:31Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ed/6dcf32465119b22d92b120fdc2f9e5ed.pdf |
| spelling | umjimathkievua-article-18822019-12-05T09:30:37Z Almost Menger property in bitopological spaces Майже Менгерова властивiсть у бiтопологiчних просторах Eysen, A. F. Özçağ, E. Ейсен, А. Е. Озчаг, Е. We introduce the notion of almost Menger property in bitopological spaces. We give some characterizations in terms of $(i, j)$-regular open sets and almost continuous surjection. We also investigate the notion of almost $\gamma$ -set in the bitopological context. Введено поняття майже менгерової властивостi в бiтопологiчних просторах, наведено деякi характеристики в термiнах $(i, j)$-регулярних вiдкритих множин i майже неперервної сюр’єкцiї та вивчено поняття майже $\gamma$ -множини в бiтопологiчному контекстi. Institute of Mathematics, NAS of Ukraine 2016-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1882 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 6 (2016); 835-841 Український математичний журнал; Том 68 № 6 (2016); 835-841 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1882/864 Copyright (c) 2016 Eysen A. F.; Özçağ E. |
| spellingShingle | Eysen, A. F. Özçağ, E. Ейсен, А. Е. Озчаг, Е. Almost Menger property in bitopological spaces |
| title | Almost Menger property in bitopological spaces |
| title_alt | Майже Менгерова властивiсть у бiтопологiчних просторах |
| title_full | Almost Menger property in bitopological spaces |
| title_fullStr | Almost Menger property in bitopological spaces |
| title_full_unstemmed | Almost Menger property in bitopological spaces |
| title_short | Almost Menger property in bitopological spaces |
| title_sort | almost menger property in bitopological spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1882 |
| work_keys_str_mv | AT eysenaf almostmengerpropertyinbitopologicalspaces AT ozcage almostmengerpropertyinbitopologicalspaces AT ejsenae almostmengerpropertyinbitopologicalspaces AT ozčage almostmengerpropertyinbitopologicalspaces AT eysenaf majžemengerovavlastivistʹubitopologičnihprostorah AT ozcage majžemengerovavlastivistʹubitopologičnihprostorah AT ejsenae majžemengerovavlastivistʹubitopologičnihprostorah AT ozčage majžemengerovavlastivistʹubitopologičnihprostorah |