Abstract
In this paper, we introduce a new type of mixed fuzzy topological space. We define countability on mixed fuzzy topological spaces. We investigate its different quasi type properties.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
1 Introduction
The different notions of topological space has been generalized and expanded in many ways in the last half century. The introduction of the notions of bitopological space and mixed topological space are two major developments in topological spaces. Bitopological spaces has recently been studied by Ganguly and Singha (1984), Tripathy and Sarma (2011b, 2012) and many others. For a detailed account on Bitopological spaces, one may refer to Hussain (1966). Mixed topology is a technique of mixing two topologies on a set to get a third topology. The works on mixed topology is due to Buck (1952), Cooper (1971), Wiweger (1961), Das and Baishya (1995) and many others.
Mixed topology lies in the theory of strict topology of the spaces of continuous functions on locally compact spaces. The study of classical mixed topology is very old. Buck (1952) introduced and investigated the concept of strict topology. This has been applied in the study of spectral synthesis, spaces of bounded holomorphic functions and multipliers of Banach algebras. Conway (1967) studied strict topology and compactness in the space of measures. Cooper (1971) applied strict topology in investigating some results in function spaces. Orlicz (1955) has applied mixed topology in summability theory. It has been applied for the study in non-locally compact spaces, interpolation theorems for analytic functions and Schauder decomposition by Subramanian (1972) and Gupta et al. (1983). There are many other applications of mixed topology.
Zadeh introduced the concept of fuzzy sets in the year 1965. Since then the notion of fuzziness has been applied for the study in all the branches of science and technology. It has been applied for studying different classes of sequences of fuzzy numbers by Tripathy and Baruah (2010), Tripathy and Borgohain (2008), Tripathy and Dutta (2010), Tripathy and Sarma (2011a) and many workers on sequence spaces in the recent years. The notion of fuzziness has been applied in topology and different notions of fuzzy topological spaces have been introduced and investigated by many researches in topological spaces. Different properties of fuzzy topological spaces have been investigated by Arya and Singal (2001a, b), Chang (1968), Das and Baishya (1995), Ganster et al. (2005), Ganguly and Singha (1984), Ghanim et al (1984), Katsaras and Liu (1977), Petricevic (1998), Srivastava et al. (1981), Warren (1978), Wong (1974a, b) and many others. Recently mixed fuzzy topological spaces are being investigated from different aspect by Das and Baishya (1995).
2 Preliminaries
Let X be a non-empty set and I, the unit interval [0, 1]. A fuzzy set A in X is characterized by a function \( \mu_{A} :X \to I \) where μ A is called the membership function of A and μ A (x) representing the membership grade of x in A. The empty fuzzy set is defined by \( \mu_{\Upphi } (x) = 0 \) for all x ∈ X. In addition, X can be regarded as a fuzzy set in itself defined by \( \mu_{X} (x) = 1 \) for all x ∈ X. Further, an ordinary subset A of X can also be regarded as a fuzzy set in X if its membership function is taken as usual characteristic function of A that is \( \mu_{A} (x) = 1 \) for all x ∈ A and \( \mu_{A} (x) = 0 \) for all x ∈ X − A. Two fuzzy set A and B are said to be equal if \( \mu_{A} = \mu_{B} . \) A fuzzy set A is said to be contained in a fuzzy set B, written as \( A \subseteq B \), if \( \mu_{A} \le \mu_{B} \). Complement of a fuzzy set A in X is a fuzzy set \( A^{\prime} \) in X defined by \( \mu_{A} = 1 - \mu_{{A^{c} }} \). We write \( A^{c} = {\text{co}}A. \) Union and intersection of a collection {A i : i ∈ N} of fuzzy sets in X, are written as \( \mathop \cup\nolimits_{i \in N} A_{i} \;{\text{and}}\;\mathop \cap \nolimits_{i \in N} A_{i} \), respectively. The membership functions are defined as follows:
A fuzzy topology τ on X is a collection of fuzzy sets in X such that Φ, X ∈ τ; If A i ∈ τ, i ∈ N then \( \mathop \cup_{i \in N} A_{i} \in \tau \) and if A, B ∈ τ then \( A \cap B \in \,\tau . \) The pair (X, τ) is called a fuzzy topological space (fts). Members of τ are called open fuzzy sets and the complement of an open fuzzy set is called a closed fuzzy set. If (X, τ) is a fts then, the closure and interior of a fuzzy set A in X, denoted by cl A and int A, respectively, are defined as cl A = \( \cap \){B: B is a closed fuzzy set in X and A \( \subseteq \) B} and int A = \( \cup \){V : V is an open fuzzy set in X and V \( \subseteq \) A}. Clearly, cl A (respectively, int A) is the smallest (respectively, largest) closed (respectively, open) fuzzy set in X containing (respectively, contained in) A. If there are more than one topologies on X, then the closure and interior of A with respect to a fuzzy topology τ on X will be denoted by τ-cl A and τ-int A.
Definition 2.1
A collection \( \fancyscript{B} \) of open fuzzy sets in a fts X is said to be an open base for X if every open fuzzy set in X is a union of members of \( \fancyscript{B} \).
Definition 2.2
If A is a fuzzy set in X and B is a fuzzy set in Y, then A × B is a fuzzy set in X × Y defined by \( \mu_{AxB} (x,y) = \min \{ \mu_{A} (x),\mu_{B} (x)\} \) for all x ∈ X and for all y ∈ Y.
Definition 2.3
Let f be a function from X into Y. Then for each fuzzy set B in Y, the inverse image of B under f, written as f −1[B], is a fuzzy set in X defined by \( \mu_{{f^{ - 1} [B]}} (x) = \mu_{B} (f(x)) \) for all x ∈ X.
Definition 2.4
A fuzzy set A in a fuzzy topological space (X, τ) is called a neighbourhood of a point x ∈ X if and only if there exists B ∈ τ such that B \( \subseteq \) A and A(x) = B(x) > 0.
Definition 2.5
A fuzzy point x α is said to be quasi-coincident with A, denoted by x α q A, if and only if α + A(x) > 1 or α > (A(x))c.
Definition 2.6
A fuzzy set A is said to be quasi-coincident with B, denoted by AqB, if and only if there exists a x ∈ X such that A(x) + B(x) > 1.
It is clear from the above definition, if A and B are quasi-coincident at x both A(x) and B(x) are not zero at x and hence A and B intersect at x.
Definition 2.7
A fuzzy set A in a fts (X, τ) is called a quasi-neighbourhood of x λ if and only if A 1 ∈ τ such that \( \overline{{A_{1} }} \subseteq A \) and x λ qA1. The family of all Q-neighbourhood of x λ is called the system of Q-neighbourhood of x λ. Intersection of two quasi-neighbourhood of x λ is a quasi-neighbourhood.
3 Main results
Das and Baishya (1995) have defined the mixed fuzzy topology with respect to a fuzzy point x λ. Recently, Ali (2009) has defined the mixed fuzzy topology on taking the neighbourhood of a fuzzy point. In this paper, we have considered a fuzzy set. Thus, our definition of mixed fuzzy topology generalizes the existing type of construction of mixed fuzzy topology.
We introduce a new type of mixed fuzzy topological space in this section.
Theorem 3.1
Let (X, τ1) and (X, τ2) be two fuzzy topological spaces. Consider the collection of fuzzy sets τ1(τ2) = {A ∈ I X : For any fuzzy set B in X with AqB, there exists τ 2 -open set A 1 such that A 1 qB and τ 1 -closure \( \overline{{A_{1} }} \subseteq A \)}. Then this family of fuzzy sets will form a topology on X and this topology we call a mixed fuzzy topology on X.
Proof
First, we verify that this family will form a topology on X.
-
(T 1) \( \overline{0} \) ∈ τ1(τ2) since \( \overline{0} \) is not quasi-coincident with any fuzzy set A in X, and therefore, there does not arise any questions of violation of the condition of being member of τ1(τ2). \( \overline{1} \in \tau_{1} \left( {\tau_{2} } \right) \) since for any fuzzy set \( A,\;Aq\overline{1} \) and there exists τ2-open set A 1 with \( A_{1} q\overline{1} \) and τ1-closure \( \overline{{A_{1} }} \subseteq \overline{1} .\)
-
(T 2) To show that intersection of any two members of τ1(τ2) is again a member of τ1(τ2).
Let A 1 , A 2 ∈ τ1(τ2). We show that \( A_{1} \cap A_{2} \) ∈ τ1(τ2).
Since A 1 ∈ τ1(τ2), so for any fuzzy set A in X with AqA 1 there exists τ2-open set B 1 such that B 1 qA and τ1-closure \( \overline{{B_{1} }} \subseteq A_{1} . \)
Also, A 2 ∈ τ1(τ2) so for any fuzzy set A in X with A q A 2 there exists τ2-open set B 2 such that B 2 qA and τ1-closure \( \overline{{B_{2} }} \subseteq A_{2} . \)
Now, B 1 , B 2 are τ2-open set implies \( B_{1} \cap B_{2} \in \tau_{1} \left( {\tau_{2} } \right). \)
We have \( \overline{{B_{1} \cap B_{2} }} \subseteq \overline{{B_{1} }} \cap \overline{{B_{2} }} \subseteq A_{1} \cap A_{2} , \) where \( \overline{{B_{1} \cap B_{2} }} \) is the τ1-closure of \( B_{1} \cap B_{2} \).
Thus, we get a τ2-open set \( B_{1} \cap B_{2} \) such that \( \left( {B_{1} \cap B_{2} } \right) \) qA and hence \( \overline{{B_{1} \cap B_{2} }} \subseteq A_{1} \cap A_{2} . \)
Therefore, we have \( A_{1} \cap A_{2} \in \tau_{1} \left( {\tau_{2} } \right). \)
-
(T 3) Let \(\{A_\lambda : \lambda \in \wedge\}\) be a family of open sets. We need to show that \( \mathop \vee \limits_{\lambda \in \wedge } A_{\lambda } \in \tau_{1} \left( {\tau_{2} } \right). \)
Since A λ ∈ τ1(τ2) \(\forall{\lambda} \in {\wedge}, \) thus for any fuzzy set A such that AqA λ, for all \({\lambda} \in {\wedge}. \)
$$ \Rightarrow \hbox{There exists } x \in X\hbox{ such that } A(x) + A_\lambda(x) > 1. $$$$ \Rightarrow A\left( x \right) \, + \mathop {{\rm Max}}\limits_{\lambda \in \wedge } \left\{ {A_{\lambda } \left( x \right) \, } \right\} > 1. $$$$ \Rightarrow A\left( x \right) \, + \mathop \vee \limits_{\lambda \in \wedge } A_{\lambda } \left( x \right) \, > \, 1. $$$$ \Rightarrow A \, q(\mathop \vee \limits_{\lambda \in \wedge } A_{\lambda } ). $$Since A λ ∈ τ1(τ2) and AqA λ, for any fuzzy set A, so there exists τ2-open fuzzy open set B λ such that B λ qA and τ1-closure cl(B λ) ≤ Aλ.
But we know that arbitrary union of member of τ2 is also a member of τ2 and so \( \mathop \vee \nolimits_{\lambda \in \wedge } B_{\lambda } \in \tau_{2} \,{\text{and}} \,(\mathop \vee \nolimits_{\lambda \in \wedge } B_{\lambda } )qA. \) On considering the τ1-closure, we have
$$ \vee (cl\left( {B_{\lambda } } \right) \le cl(\mathop \vee \limits_{\lambda \in \wedge } B_{\lambda } ) \le \;\mathop \vee \limits_{\lambda \in \wedge } A_{\lambda } \Rightarrow \;\mathop \vee \limits_{\lambda \in \wedge } A_{\lambda } \in \tau_{1} \left( {\tau_{2} } \right). $$
Therefore, this collection τ1(τ2) of fuzzy subsets of X is a fuzzy topology on X. We call (X, τ1(τ2)) a mixed fuzzy topological space.
Remark 3.1
In order to have a clear idea about the mixed fuzzy topology τ1(τ2) defined in Theorem 3.1, we consider the following example.
Example 3.1
Consider the set of real numbers R and let us consider the collection of fuzzy sets \( \{ {\overline{0} ,\,\overline{1} ,\,B_{{{\text{x}},{\text{n}}}} }\} \) in R, where B x,n is defined by \( B_{x,n} \left( t \right) = \left\{ {\begin{array}{ll} {nt - nx + 1,}& {x - \frac{1}{n} \le t \le x;} \\ {nt - nx - 1,} & {x \le t \le x + \frac{1}{n};} \\ 0, & {\rm otherwise.}\\ \end{array} } \right. \)
Then the collection \( \tau_{ 1} = \left\{ {\overline{0} ,\,\overline{1} ,\,B_{x,n} } \right\} \) will form a fuzzy topology on R.
Consider the fuzzy set in R defined by
Then the collection \( \tau_{2} = \left\{ {\overline{0} ,\,\overline{1} ,\,A_{{{\text{x}},{\text{n}}}} } \right\} \) will form a fuzzy topology on R.
Now, consider the collection τ1(τ2) = {A ∈ I R: For any B ∈ I R with AqB there exists τ2-open set A x,n such that A x,n qB and τ1-closure \( \left. {\overline{{A_{x,n} }} \subseteq A} \right\}. \)
First, we show that the collection of fuzzy sets τ1(τ2) will form a mixed fuzzy topology on R.
\( \overline{0} \in \tau_{1} \left( {\tau_{2} } \right). \) Let B ∈ I R, \( \overline{0} \) is not quasi-coincident with B. since we cannot find a τ2-open set \( A_{{x_{0} ,n}} \) such that \( A_{{x_{0} ,n}} \) qB and τ1-closure \( \overline{{A_{{x_{0} ,n}} }} \subseteq \overline{0} . \)
Also, \( \overline{1} \, \in \,I^{R} . \) Let \( {\text{B}} \ne \overline{0} \, \in \, I^{R} , \) we have \( Bq\overline{1} . \)
Let the fuzzy set B be defined by
Obviously, \( Bq\,\overline{1} ,\,A_{{x_{0} ,\frac{1}{3}}} qB \;( {{\hbox{Since }} A_{{x_{0} ,\frac{1}{3}}} ( t ) \, + B( t ) > 1} ) \) and τ1-closure \( \overline{{A_{{x_{0} ,\frac{1}{3}}} }} \subseteq \overline{1} . \)
Thus, for any B ∈ I R with \( \overline{1} qB \), there exists τ2-open set \( A_{{x_{0} ,\frac{1}{3}}} \) such that \( A_{{x_{0} ,\frac{1}{3}}} qB \) and the τ1-closure \( \overline{{A_{{x_{0} ,\frac{1}{3}}} }} \subseteq \overline{1} . \) Hence \( \overline{1} \, \in \,\tau_{ 1} \left( {\tau_{ 2} } \right). \)
Next, let A 1, A 2 ∈ τ1(τ2), We have to show that \( A_{1} \wedge A_{2} \, \in \tau_{1} \left( {\tau_{2} } \right). \)
Let B be any fuzzy set in R such that B q \( \left( {A_{1} \wedge A_{2} } \right). \)
Since, A 1, A 2 ∈ τ1(τ2), so there exists τ2-open sets \( A_{{x_{1} ,n}} ,\;A_{{x_{2} ,n}} \) such that \( A_{{x_{1} ,n}} qB\;{\text{and}}\;A_{{x_{2} ,n}} qB \) and the τ1-closure \( \overline{{A_{{x_{1} ,n}} }} \subseteq A_{1} \;{\text{and}}\;\overline{{A_{{x_{2} ,n}} }} \subseteq A_{2} . \)
Now, \( A_{{x_{1} ,n}} qB\;{\text{and}}\;A_{{x_{2} ,n}} qB. \)
and τ2 is a fuzzy topology, therefore, \( A_{{x_{1} ,n}} \wedge A_{{x_{2} ,n}} \in \tau_{2} . \)
We have \( \left( {A_{{x_{1} ,n}} \wedge A_{{x_{2} ,n}} } \right)\left( t \right) + B\left( t \right) = {\text{Min}}\{ A_{{x_{1} ,n}} \left( t \right),\;A_{{x_{2} ,n}} \left( t \right)\} + B\left( t \right) > 1. \)
Thus, for any fuzzy set B in R with \( Bq(A_{1} \wedge A_{2} ), \) there exists τ2-open set \( A_{{x_{1} ,n}} \wedge A_{{x_{2} ,n}} \)such that \( \left( {A_{{x_{2} ,n}} \wedge A_{{x_{2} ,n}} } \right)qB \) and the τ1-closure \( \overline{{A_{{x_{1} ,n}} \wedge A_{{x_{2} ,n}} }} = \overline{{A_{{x_{1} ,n}} }} \wedge \overline{{A_{{x_{2} ,n}} }} \subseteq A_{1} \wedge A_{2} . \)
Therefore, \( A_{1} \wedge A_{2} \in \tau_{ 1} \left( {\tau_{ 2} } \right). \)
Finally, let us consider a collection {A i : i ∈ ∆} of fuzzy sets in τ1(τ2).
We have to show that \( \mathop \vee \nolimits_{i \in \Updelta } A_{\text{i}} \in \tau_{1} \left( {\tau_{2} } \right). \)
Let B be any fuzzy set in R such that \( Bq ( {\mathop \vee \nolimits_{i \in \Updelta } A_{\text{i}} } ). \)
Since A i ∈ τ1(τ2) and BqA i , for some i, so there exists τ2-open set \( A_{{x_{i} ,n}} \) such that Bq \( A_{{x_{i} ,n}} \) and τ1-closure \( \overline{{A_{{x_{i} ,n}} }} \subseteq A_{i} . \)
Let Bq \( A_{{x_{i} ,n}} \) for some i ∈ ∆
Since \( A_{{x_{i} ,n}} \) are open in \( \tau_{2} \Rightarrow \mathop \vee \nolimits_{i \in \Updelta } A_{{x_{i} ,n}} \) is open in τ2 and τ1-closure \( \overline{{ \vee A_{{x_{i} ,n}} }} \subseteq \vee A_{\text{i}} . \)
Therefore, if \( Bq \vee A_{\text{i}} \) then there exist τ2-open set \( \mathop \vee \nolimits_{i \in \Updelta } A_{{x_{i} ,n}} \) such that Bq \( A_{{x_{i} ,n}} \) and τ1-closure \( \overline{{ \vee A_{{x_{i} ,n}} }} \subseteq \vee A_{\text{i}} . \)
Hence \( \mathop \vee \nolimits_{i \in \Updelta } \) A i ∈ τ1(τ2) and so τ1(τ2) is a mixed fuzzy topology on R.
4 Countability on mixed fuzzy topological spaces
First, we list some known definitions, those will be used for establishing the results of this section.
Definition 4.1
Let U CQ be a family of Q-neighbourhood of a fuzzy point \( x_{\lambda } \) in X. A subfamily B CQ of U CQ is said to be a Q-neighbourhood base of U CQ if for any A ∈ U CQ there exists B ∈ B CQ such that B < A.
Definition 4.2
A fuzzy topological space (X,δ) is said to be Q-first countable space if and only if every fuzzy point in X has countable Q-neighbourhood base.
Definition 4.3
Let U C be a family of neighbourhoods of a fuzzy point \( x_{\lambda } \) in X. A subfamily B C of U C is said to be a neighbourhood base of U C , if for any A ∈ U C , there exists B ∈ U C such that B < A.
Definition 4.4
A fuzzy topological space (X,δ) is said to be first countable space if and only if every fuzzy point in X has a countable neighbourhood base.
The following definition is an alternative to the Definition 4.4.
Definition 4.5
Let (X,δ) be a fuzzy topological space. Then X is said to be a first countable space, if for each fuzzy point \( x_{\lambda } (0 \le \lambda \le 1) \) there exists a countable class of fuzzy open sets \( B_{{x_{\alpha } }} \) such that λ < U(x), for all \( U \in B_{{x_{\alpha } }} \) and if λ < V(x) for some fuzzy open set V then there exists \( W \in B_{{x_{\alpha } }} \) such that \( W \subseteq V. \)
Definition 4.6
A fuzzy topological space (X,τ) is said to be C II if there exists a countable base for τ.
We now introduce the following definitions:
Definition 4.7
Let \( \left( {X,\tau_{1} (\tau_{2} )} \right) \) be a mixed fuzzy topological space. Let \( U_{\lambda } \) be a family of neighbourhood of a fuzzy point \( x_{\lambda } \) in X. A subfamily \( B_{\lambda } \;{\text{of}}\;U_{\lambda } \) is said to be a neighbourhood base of \( U_{\lambda } \) if for any \( A \in U_{\lambda } \) there exists \( B \in B_{\lambda } \) such that B < A.
Definition 4.8
Let \( \left( {X,\tau_{1} (\tau_{2} )} \right) \) be a mixed fuzzy topological space. Then X is said to be first countable space if every fuzzy point in X has a countable neighbourhood base.
Definition 4.9
Let \( \left( {X,\tau_{1} (\tau_{2} )} \right) \) be a mixed fuzzy topological space. Then X is said to be Q-first countable space (i.e. Q–C I space) if every fuzzy point in X has a Q-neighbourhood base.
Theorem 4.1
Let \( \left( {X,\tau_{1} (\tau_{2} )} \right) \) be a C I -space. Then it is a Q–C I space.
Proof
Let e be any fuzzy point in X. Consider a sequence \( \{ \mu_{n} \}_{n \in N} \) in (1 − λ,1] converging to 1 − λ and let \( x_{{\mu_{n} }} = e_{n} \).
Since X is a C I -space for each n ∈ N, there exists a countable open neighbourhood base {B n} of e n . We have for each member B of {B n }, B(x) ≥ μ n > 1 − λ.
Hence B is a Q-neighbourhood of e. Thus, the collection {B n} is a family of open Q-neighbourhoods of “e” and hence this family will be a countable family of Q-neighbourhoods of “e”.
Let A be an arbitrary Q-neighbourhood of “e”. Hence A(x) > 1 − λ. Since \( \mu_{n} > 1 - \lambda , \) so there exists m ∈ N such that \( A\left( {\text{x}} \right) \ge \mu_{m} 1 - \lambda \Rightarrow e_{m} \in A \) and A is an open neighbourhood of “e m”.
Thus, there exists a member B ∈ {B n } such that B < A and B(x) > μ m > 1 − λ and so B is a Q-neighbourhood base of “e”.
Hence \( \left( {X,\tau_{1} (\tau_{2} )} \right) \) is a Q–C I space.
Proposition 4.2
If \( \left( {X,\tau_{1} (\tau_{2} )} \right) \) is C II , then it is also Q–C I .
Proof
Let \( \left( {X,\tau_{1} (\tau_{2} )} \right) \) be C II, so there exists a countable base for \( \tau_{1} (\tau_{2} ). \)
Let \( \fancyscript{B} \) be a countable base for \( \tau_{1} (\tau_{2} ). \) We have to show that \( \left( {X,\tau_{1} (\tau_{2} )} \right) \) is Q–C I. It is sufficient to establish that every fuzzy point in X has a countable Q-neighbourhood base.
Let e be a fuzzy point in \( \left( {X,\tau_{1} (\tau_{2} )} \right). \)
Let A be any subset of X satisfying Aqe. Hence there exists B ∈ \( \fancyscript{B} \) such that eqB and B < A. Therefore, B is a Q-neighbourhood of the fuzzy point “e”.
Let U be the family of all those members B ∈ \( \fancyscript{B} \). Clearly, this collection is a countable collection of Q-nbd. of e. That is, the fuzzy point e has a countable Q-neighbourhood base and hence \( \left( {X,\tau_{1} (\tau_{2} )} \right) \) is a Q–C I space.
Proposition 4.3
Let τ 1 and τ 2 be two fuzzy topologies for X and if the mixed fuzzy topology τ 1 (τ 2 ) is Q-first countable, then τ 2 is also Q-first countable.
Proof
Let x λ be an arbitrary fuzzy point in X.
Since τ1(τ2) is Q–C I space, therefore there exists a Q-neighbourhood base for every fuzzy point x λ . Let A ∈ B Q , where B Q is the countable collection of τ1(τ2) Q-neighbourhood base at x λ . Then A is τ1(τ2) Q-nbd. of x λ.
⇒ There exists B ∈ τ1(τ2) such that \( B \subseteq A \) and x λ qB.
We know that \( \tau_{1} \left( {\tau_{2} } \right) \subseteq \tau_{2.} \)
Therefore, B ∈ τ1(τ2) ⇒ B ∈ τ2 and \( B \subseteq A, \) x λ qB.
i.e. A is also a τ2–Q–nbd. of x λ.
Thus, every member A ∈ B Q is τ2–Q-neighbourhood of x λ .
⇒ B Q is also τ2-Q-neighbourhood base at x λ .
Hence, τ2 is also Q–C I space.
References
Ali T (2009) Fuzzy topological dynamical systems. J Math Res 1(2):199–206
Arya SP, Singal N (2001a) Fuzzy subweakly continuous functions. Math Stud 70(1–4):231–240
Arya SP, Singal N (2001b) Fuzzy subweakly α-continuous functions. Math Stud 70(1–4):241–246
Buck RC (1952) Operator algebras and dual spaces. Proc Am Math Soc 3:681–687
Chang CL (1968) Fuzzy topological spaces. J Math Anal Appl 24:182–190
Conway JB (1967) The strict topology and compactness in the space of measure. Trans Am Math Soc 126:476–486
Cooper JB (1971) The strict topology and spaces with mixed topologies. Proc Am Math Soc 30:583–592
Das NR, Baishya PC (1995) Mixed Fuzzy Topological Spaces. J Fuzzy Math 3(4):777–784
Ganguly S, Singha D (1984) Mixed topology for a bi-topological spaces. Bull Cal Math Soc 76:304–314
Ganster M, Georgiou DN, Jafari S, Mosokhoa S (2005) On some application of fuzzy points. Appl Gen Topol 6(2):119–133
Ghanim MH, Kerre EE, Mashhour AS (1984) Separation axioms, subspaces and sums in fuzzy topology. J Math Anal Appl 102:189–202
Gupta M, Kamthan PK, Das NR (1983) Bi-locally convex spaces and Schauder decomposition. Ann Math Pure Appl XXXIII(iv):267–284
Hussain T (1966) Introduction to topological groups. W.B. Saunders, Philadelphia
Katsaras AK, Liu DB (1977) Fuzzy vector spaces and fuzzy topological vector spaces. J Math Anal Appl 58:135–146
Orlicz W (1955) Linear operator in saks space. Studia Math 15:1–25
Petricevic Z (1998) On s-closed and extremally disconnected fuzzy topological spaces. Matematички Bechик 50(1–2):37–45
Srivastava R, Lal SN, Srivastava AK (1981) Fuzzy Hausdorff topological spaces. J Math Anal Appl 81:491–506
Subramanian PK (1972) Two-norm spaces and decompositions of Banach spaces, I. Studia Math T XLIII:179–194
Tripathy BC, Baruah A (2010) Lacunary statistically convergent and lacunary strongly convergent generalized difference sequences of fuzzy real numbers. Kyungpook Math J 50(4):565–574
Tripathy BC, Borgogain S (2008) The sequence space m(M,ϕ, Δ n m , p)F. Math Modell Anal 13(4):577–586
Tripathy BC, Dutta AJ (2010) Bounded variation double sequence space of fuzzy real numbers. Comput Math Appl 59(2):1031–1037
Tripathy BC, Sarma B (2011a) Double sequence spaces of fuzzy numbers defined by Orlicz function. Acta Math Sci 31B(1):134–140
Tripathy BC, Sarma DJ (2011b) On b-locally open sets in bitopological spaces. Kyungpook Math J 51(4):429–433
Tripathy BC, Sarma DJ (2012) On pairwise b-locally open and pairwise b-locally closed functions in bitopological spaces. Tamkang J Math 43(4) (accepted)
Warren RH (1978) Neighbourhood bases and continuity in fuzzy topological spaces. Rocky Mt J Math 8:459–470
Wiweger A (1961) Linear spaces with mixed topology. Studia Math. 20:47–68
Wong CK (1974a) Fuzzy topology: product and quotient theorems. J Math Anal Appl 45:512–521
Wong CK (1974b) Fuzzy points and local properties of fuzzy topology. J Math Anal Appl 46:512–521
Acknowledgments
The authors thank the referees and the Editor for their comments those improved the presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tripathy, B.C., Ray, G.C. On mixed fuzzy topological spaces and countability. Soft Comput 16, 1691–1695 (2012). https://doi.org/10.1007/s00500-012-0853-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-012-0853-1