On Pairwise C-closed Bitopological Spaces
EMAN ALMUHUR
Department of Basic Science and Humanities, Faculty of Arts and Science,
Applied Science Private University,
Shafa Badran, Amman,
JORDAN
Abstract: - Ismail first proposed the concept of C-closed topological spaces in 1980, assuming that spaces
which are countably compact are closed. In 2019, Omar and Hdeib introduced the notion of pairwise C-closed
bitopological spaces. In this article, several results concerning these notions are proposed and discussed.
Many findings are summarized in relating pairwise strongly Lindelöf bitopological space and pairwise strongly
C-lindelöf.
Key-Words: - Bitopological Space, Pairwise C-closed Space, Pairwise C-Lindelöf.
Received: July 16, 2021. Revised: April 16, 2022. Accepted: May 19, 2022. Published: June 8, 2022.
1 Introduction
In 1963, Kelly [1] initiated “Bitopological Spaces”
in an article in the London Mathematical society,
thereafter a large number of articles generalized
topological concepts to bitoplogical ones. In 1980,
Ismail [2] introduced C-closed topological spaces in
which he assumed that every countably compact is
closed. Omar and Hdeib [3] introduced the concept
of pairwise C-closed bitopological spaces in 2019.
The notion of strongly Lindelöf spaces was
introduced in an article named “Strongly compact
spaces” by Mashour in 1984 where he required that
each preopen cover of the space to have a countable
subcover.
Omar and Hdeib [3] called a bitopological space
a pairwise countably compact space if the
countably open cover of has a finite subcover.
Also, they introduced the notion of pairwise C-
closed spaces where every countably compact
subset of a bitopological space is
closed and every countably compact subset
of is closed [4].
2 Preliminaries
The bitopological spaces and
(simply and ) are bitopological spaces on which
no separation axioms are required unless clearly
indicated throughout this work.
For the bitopological space :
i. A cover
of the bitoplogical space is
pairwise open cover if
covers .
ii. is pairwise countably compact if the countably
pairwise open cover of has finite subcover.
iii. is called p-Hausdorff [1] if in ,
and
.
vi. is regular with respect to if and
closed subset not containing , a
open subset of and which is a open
subset of and disjoint from such that
and
v. is p-regular [1] if is regular with respect to
and is regular with respect to .
3 Pairwise C-closed Bitopological
Spaces
Definition 1: If is a bitopological space,
then:
i. A subset of an preopen (resp.
preclosed) [5] if
(resp.
If is preopen and
preopen, so is pairwise preopen. A
pairwise preopen complement subset is pairwise
preclosed.
ii. A bitopological space is said to be
pairwise strongly C-closed [6] if a
countably compact subset of is
preclosed and countably compact
subset of is preclosed.
iii. A bitopological space is a pairwise
C- Lindelöf [7] if every preclosed
subset is Lindelöf and every
preclosed subset is Lindelöf.
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iv. A bitopological space is pairwise
strongly C- Lindelöf [4] if for every
preclosed set , there exists an
preopen cover of contains a
countable subfamily such that
is covered by
where the preclosure of a subset is the
smallest preclosed set containing it
Lemma1: A pairwise strongly Lindelöf
bitopological space is pairwise strongly C- Lindelöf.
Proof: Assume that is pairwise strongly
Lindelöf space and let be a cover of
pairwise preopen subsets, then the open cover
consists of pairwise open subsets since
each preopen subset is open. So, a
countable subset such that
Thus is
pairwise Lindelöf.
Proposition 1: A subspace of a pairwise strongly C-
closed bitopological space is strongly C-closed.
Proposition 2: Every subspace of a pairwise
strongly Lindelöf bitopological space is pairwise
strongly Lindelöf.
Proposition 3: Every subspace of a pairwise
strongly C-Lindelöf bitopological space is pairwise
strongly C-Lindelöf.
Proposition 4: A pairwise Lindelöf bitopological
space is a pairwise C-Lindelöf.
Proof: Let be a preclosed subset of a
bitopological space , if is a
cover of consisting of open subsets of , then
is an open cover of
that admits a countable subcover
. So, is covered by a countable subcover.
Thus, is Lindelöf. Similarly, if we assume
that is a closed subset of , we will get that
is Lindelöf.
A bitopological space is called p-
Hausdorff [4] if there open
subset and a open subset disjoint from such
that and
Definition 2: In a bitoplogical space ,
is regular with respect to [8] if and each
closed subset such that , there exists a
open subset and aopen subset
disjoint from such that and
A bitopological space is p-regular [3] if
is regular with respect to and is regular with
respect to .
Definition 3: A subset of is said to be
regular open (resp. regular closed) if
(resp.
). If is regular open and
regular open, then is pairwise regular
open. The complement of a pairwise regular open is
also pairwise regular closed.
Proposition 5: If the bitopological space
is p-regular pairwise C- Lindelöf, then is pairwise
Lindelöf.
Proof: Let be a p-regular pairwise C-
Lindelöf, that is not pairwise Lindelöf. Let
be an open cover of that has no
countable subcover, but is C- Lindelöf, so a
subcover consisting of pairwise preclosed subsets of
it admits a countable subcover , i.e there exists a
countable subset such that
where
is pairwise preclosed subset of .
Hence, is pairwise Lindelöf.
Proposition 6: Every pairwise C- Lindelöf p-regular
bitopological space is pairwise strongly Lindelöf.
Proposition 7: A pairwise strongly C- Lindelöf p-
regular is a pairwise strongly Lindelöf.
A bitopological space is said to be
pairwise sequential [1] if every non closed
subset of contains a sequence that converges to
a point in
Proposition 8: The p-Hausdorff pairwise sequential
bitopological space is pairwise strongly C-closed.
Proof: Suppose that is a countably compact
subset of which is not preclosed, then
such that
Assume that , then is countably
compact. Since is a pairwise sequential, in
a sequence in such that converges to
that is does not have cluster
points in which is a contradiction.
Proposition 9: A pairwise strongly C-Lindelöf p-
regular bitopological space is pairwise Lindelöf.
Corollary 1: A pairwise strongly Lindelöf
bitopological space is a pairwise C-Lindelöf.
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Proposition 10:[1] Let be a pairwise p-
Hausdorff bitopological space, let be a
convergent sequence in , then it has exactly one
limit point.
Proposition 11: Let be a p-Hausdorff, if
every pairwise countably compact subset of is
pairwise sequential, then is pairwise strongly C-
closed.
Proof: Suppose that is a countably compact
subset of and that is not preclosed. If
, and , then is
countably compact. is not closed in ,
but is sequential, then there exists a sequence
in such that . Hence,
is a sequence in that has no limit points
in which is a contradiction.
Consider the bitopological spaces and
, the function
is pairwise continuous provided that it is continuous
both as a map from to and as map
from to .
Proposition 12: If a p-Hausdoff bitopological space
admits a pairwise continuous surjective
mapping into a pairwise C-closed space ,
then is pairwise C-closed.
Proof: Suppose the bitopological space is
p-Housdorff a pairwise and that is a
pairwise C-closed space, let
be a pairwise continuous surjective
function, if is a countably compact subset of
, then its image under
is
a countably compact subset of , similarly for
in under . Now,
since is a pairwise C-closed space, is
closed in Thus,
is
closed in
Proposition 13: If is a p-regular
bitopological space, then if every point has a
pairwise C-closed neighbourhood, then is
pairwise C-closed.
Proof. Suppose that be a pairwise countably
compact subset of . For any point in , there
exists a p-open subset of containing . Since X
is p-regular, there exists a p-open subset of such
that and Now, is a
pairwise countably compact subset of ,
, hence is a neighbourhood of .
Proposition 14: If is a p-regular pairwise
countably compact, then each pairwise
subset is
pairwise closed in .
Proof: Suppose that
be a family of
pairwise closed subsets of , and assume that
is not pairwise closed, if and
, then is pairwise countably compact
first countable at since is pairwise . There is a
sequence in converging to , hence this
sequence has no cluster points in which
contradicts the assumption.
Proposition 15: If is a p-regular
bitopological space and each of its pairwise
countably compact subsets is an
, then is a
pairwise C-closed space.
Proof: By proposition 14, we get the result.
Proposition 16: If is a p-regular
bitopological space and is the countable union of
its pairwise C-closed subspaces, then is pairwise
C-closed.
Proposition 17: If is a p-regular
bitopological space and its points are then is
pairwise C-closed.
Proof: Suppose that is a pairwise countably
compact subset of and that is not pairwise
closed. If ,
and
is a first countable pairwise countably compact
subset of . Consequently there exists a sequence
converging to , that is is not a cluster point
which is a contradiction. Thus assumed result is
hold.
Proposition 18: If is a p-Hausdorff
bitopological space and each pairwise countably
compact subset of is sequential, then is pairwise
C-closed [9].
Corollary 2: Each pairwise countably compact
subset of the bitopological space is
sequential and is not sequential.
Corollary 3: Each subspace of the pairwise C-
closed space is pairwise C-closed.
Lemma 2: If is a pairwise countably
compact C-closed space, then its cardinality is less
than or equal to where the density of is
denoted by .
Proof: Consider the subset of , if
, let be a limit point of and
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, then l .
If is a dense subset of and , then
and hence . If
, then is pairwise countably compact
and and is a pairwise dense subset
of , thus .
4 Conclusion
Every pairwise strongly Lindelöf bitopological
space is pairwise strongly C- Lindelöf and each
subspace of a pairwise strongly C-closed
bitopological space is strongly C-closed. For the
subspaces of a pairwise strongly Lindelöf
bitopological spaces, each subspace is pairwise
strongly Lindelöf and the subspace of a pairwise
strongly C-Lindelöf bitopological space is pairwise
strongly C-Lindelöf. For the pairwise Lindelöf
bitopological space, they are pairwise C-Lindelöf
spaces. Pairwise C- Lindelöf p-regular bitopological
space is pairwise strongly Lindelöf and strongly C-
Lindelöf p-regular spaces are pairwise strongly
Lindelöf.
References:
[1] Kelly J. C., Bitopological spaces, London Math.
Soc. Proc. (1965), 15, pp. 71-89.
[2] Ismail M., Nyikos P. On spaces in which
countably compact sets are closed, and
hereditary properties. Topology and its
Applications,(1980), 11(3), pp. 281-292.
[3] Omar N., Hdeib H. On pairwise C-closed space
in bitopological space, Journal of Semigroup
Theory and Applications, (2019), 3, pp. 1-7.
[4] Arhangel’skii A.V. Bicompact sets and the
topology of spaces. Trudy Moskov. Mat. Qbsc.
(1965), 13, pp. 3-55.
[5] Chaber J. Conditions which imply compactness
in countably compact spaces. Bull. Acad. Polon.
Sci. Ser. Sci. Math. Astronom. Phys. (1976),
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[6] Ismail M. A note on a theorem of Arhangel’skii.
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[7] Ismail M. Generalizations of compactness and
cardinal invariants of spaces. Dissertation,
University of Pittsburgh, 1979.
[8] Olson R.S. Biquotient maps, Countably
Biquotient spaces and related topics, General
Topology and Appl. (1974), 4, pp. l-28.
[9] Omar N., Hdeib H., Almuhur E., Al-labadi M.
On P_2-C-Closed Space in Bitopological Space.
Advances in Mathematics: Scientific Journal,
(2021), 10(3), pp. 1839–1843.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The correspondence author carried out the all
propositions and their proofs.
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