BITOPOLOGICAL SPACES PDF

In this paper, we introduce a new type of closed sets in bitopological space (X, τ1, τ2), used it to construct new types of normality, and introduce new forms of. Definitions. Recall that a topological space is a set equipped with a topological structure. Well, a bitopological space is simply a set equipped. Citation. Patty, C. W. Bitopological spaces. Duke Math. J. 34 (), no. 3, doi/S

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To receive news and publication updates for Journal of Mathematics, enter your email address in the box below. Correspondence should be addressed to M. This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use, distribution, and reproduction in any medium, provided the original work is spacea cited. We are going to establish some results of – semiconnectedness and compactness in a bitopological space.

Besides, we will investigate several results in – semiconnectedness for subsets in bitopological spaces. In particular, we will discuss the relationship related to semiconnectedness between the topological spaces and bitopological space.

That is, if spacee bitopological space is – semiconnected, then the topological spaces and are -semiconnected. In addition, we introduce the result which states that a bitopological space is – semiconnected if and only if and are the only subsets of which are – semiclopen sets.

Moreover, we have proved some results in compactness also. Altogether, several results of – semiconnectedness and compactness in a bitopological space have been discussed. He introduced this concept in his journal of London Mathematical Society in He initiated his study about bitopological space by using quasimetric and its conjugate.

A quasimetric on a set is a nonnegative real valued psaces on the Cartesian product of that satisfies the following three axioms: However, the quasimetric cannot be a metric. Because the symmetric property does not hold for quasimetric. Moreover, every metric space is a topological space. But this is not true for bitopological space in general. Anyhow, bitopological spaces exist for quasimetric spaces. Maheshwari and Prasad [ 2 ] introduced semiopen sets in bitopological spaces in In bitopolotical, the notion -open sets in bitopological spaces was introduced by Banerjee [ 3 ].

After that, Khedr [ 4 ] introduced and studied about – open sets. Later, Fukutake [ 5 ] defined one kind of semiopen sets in Recently, Edward Samuel and Balan [ 6 ] established the concept – semiopen sets in bitopological spaces. Moreover, we have presented some results of – semiconnectedness in bitopological spaces in [ 8 ]. In this paper, we are going to discuss the following results: Then is also – semiconnected. Let —-and – be the interior, closure, -interior, -closure, and -semiclosure of with respect to the topologyrespectively.

Let – int and – cl are the -interior and -closure of with respect to the topologyrespectively,where and are semiregularization of andrespectively. Then the triple is called a bitopological space. Definition 2 see [ 1 ]. Let be subset of a bitopological space.

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Then is called -open, if. Complement of -open set is called -closed set. Spacea 3 see [ 9 ]. Then is called 1 -regular open, if -int -cl ; 2 -regular open, if -int -cl ; 3 -semiopen, if -cl -int ; 4 -semiclosed, if -int -cl. Definition 4 see [ 9 ]. Let be subset of bitopological space. Then, 1 is said to be – open set, if, forthere spsces -regular open set such that.

Complement of – open set is called – closed set; 2 is said to be – open set, if forthere exists -regular open set such that.

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Complement of – open set is called – closed set; 3 Collection of all – open sets and – open sets are denoted by and respectively. Definition 5 see [ 6 ]. Then, is called – semiopen set, if there exists an – open set such that -cl. Complement of – open set is called – closed set. Definition 6 see [ 6 ]. A subset is called a – semidisconnected subset of a bitopological spaceif spacees semiopen sets such thatand. Otherwise is called bitopologocal – semiconnected subset.

Definition 7 see [ 6 ]. A bitopological space is called – semiconnected space, if cannot be expressed as the union of two disjoint sets and such that – scl – scl. Suppose can be so expressed, then is called – semidisconnected space and we write and it is called – semiseparation of. A nonempty collection is called a – semiopen cover biitopological a bitopological spaceif and – – and contains at least one member of – and one member of.

A cover of a bitopological space is called – open cover ofif andand. A bitopological space is bitopolovical – compact, if every – open cover of has a finite subcover. A bitopological space is called – semicompact, if every – semiopen cover of has a zpaces subcover. Proposition 12 see [ 8 ].

bitopological space in nLab

Let be family of – semiconnected subsets of a bitopological space such that ; then is also – semiconnected. Now, assume that is not – semiconnected.

Then there exist two – semiopen sets and ifandthen let other case is similar. Now there exist such that also since and also and sincewhich shows that is – semidisconnected subset and it is a contradiction. So, is – semiconnected. If a bitopological space is – semiconnected, then is – semiconnected. Suppose is – semiconnected.

Then cannot be expressed as the union of two nonempty disjoint sets and such that – – Also is – semiopen and is – semiopen. Since andwe have every – semiopen and – semiopen are – semiopen and – semiopen, respectively.

Therefore, cannot be expressed as the union of two nonempty disjoint sets and such that is – semiopen and is – semiopen, respectively. Hence, is – semiconnected.

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Proposition 14 see [ 8 ]. If a bitopological space is – semiconnected, then the topological spaces and are -semiconnected. Since every – open set and – open set are – semiopen set and – semiopen set, respectively, So if and are -semidisconnected spaces then the bitopological space becomes -semidisconnected. But this is impossible. So and are -semiconnected spaces. Proposition 15 see [ 8 ]. A bitopological space is – semiconnected if and only if and are the only subsets of which are – semiclopen sets simultaneously semiopen and semiclosed.

Consider a – semiconnected space ; let and is – semiclopen set; then is – semidisconnected in the bitopological space, which is contradiction.

Bitopological space

So and are the only subsets of which are both – semiclopen sets. Conversely, let and be the only subsets of which are both – semiclopen sets. If the bitopological space is – semidisconnected, so there exists a – semidisconnection of the bitopological space. So and ; then and both are – semiclopen sets and each spaves them is neither nor. This is a contradiction.

Therefore, is – semiconnected. If is – semiconnected then cannot be expressed as the union of two nonempty sets with such that is – semiopen and is – semiopen. Assume that can be expressed as the union of two nonempty disjoint sets and such that is – semiopen and is – semiopen, respectively.

Sincewe have. Similarly, we can prove. This contradicts our supposition. Thus, cannot be expressed as the union of two nonempty disjoint sets and such that is – semiopen and is – semiopen.

If cannot be expressed as the union of two disjoint sets and such that is – semiopen and is – semiopen, then does not contain any nonempty proper subset which is both – semiopen and – semiclosed. Let cannot be expressed as the union of two nonempty sets with such that is – semiopen and is – semiopen. If contains a nonempty proper subset which is both – semiopen and – semiclosed. Thenwhere is – semiopen, is – semiopen, and are disjoint.

Thus, does not contain any nonempty proper subset which is both – semiopen and – semiclosed. The union of any family of – semiconnected sets with a nonempty intersection is – semiconnected. Takewhere each is – semiconnected with. Suppose that is not – semiconnected. Thenwhere and are two nonempty disjoint sets such that -. Since is – semiconnected andwe have or. Thus, is – semiconnected.

Every – semicompact space is – compact. Take to be – semicompact. Let be a pairwise open cover of.