On mixed fuzzy topological spaces and countability

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.

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:

$$ \mu_{{\mathop \cup \limits_{i \in N} A_{i} }} (x) = \sup \{ \mu_{{A_{i} }} (x):i \in N\} \quad {\text{for}}\;{\text{all }}x \in X, $$

$$ {\text{and}}\;\mu_{{\mathop \cap \limits_{i \in N} A_{i} }} (x) = \inf \{ \mu_{{A_{i} }} (x):i \in N\} \quad {\text{for all}}\,x \in X. $$

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) ≤ A_λ.

  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

$$ A_{x,n} ( t ) = \left\{ {\begin{array}{ll} 1, \\ 0, \\ \end{array} \begin{array}{ll} {x - \frac{1}{n} \le t \le x + \frac{1}{n}} \\ {\rm otherwise} \\ \end{array} } \right. $$

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

$$ B\left( t \right) = \left\{ {\begin{array}{ll} {0.5,} & { \, x - 0.5 \le t \le x + 0.5,} \\ {0,} & {{\text{otherwise}} .} \\ \end{array} } \right. $$

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 $\left( {A_{1} \wedge A_{2} } \right).$

$$ \begin{aligned} & \Rightarrow B\left( t \right) + (A_{1} \wedge A_{2} )\left( t \right) > 1,\;{\text{for some}}\;t \in R. \\ & \Rightarrow B\left( t \right) + {\text{Min}}\left\{ {A_{1} \left( t \right),A_{2} \left( t \right)} \right\} \, > 1. \\ & \Rightarrow B\left( t \right) + A_{1} \left( t \right) > 1\;{\text{and}}\;B\left( t \right) \, + A_{2} \left( t \right) > 1. \\ & \Rightarrow BqA_{1} \;{\text{and}}\;BqA_{2} . \\ \end{aligned} $$

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, $A_{{x_{1} ,n}} qB\;{\text{and}}\;A_{{x_{2} ,n}} qB.$

$$ \Rightarrow A_{{x_{1} ,n}} \left( t \right) + B\left( t \right) \, > 1\;{\text{and}}\;A_{{x_{2} ,n}} \left( t \right) + B\left( t \right) \, > 1,\;{\text{ for some}}\;t \in R. $$

(1)

and τ₂ is a fuzzy topology, therefore, $A_{{x_{1} ,n}} \wedge A_{{x_{2} ,n}} \in \tau_{2} .$

We have $\left( {A_{{x_{1} ,n}} \wedge A_{{x_{2} ,n}} } \right)\left( t \right) + B\left( t \right) = {\text{Min}}\{ A_{{x_{1} ,n}} \left( t \right),\;A_{{x_{2} ,n}} \left( t \right)\} + B\left( t \right) > 1.$

$$ \Rightarrow \left( {A_{{x_{1} ,n}} \wedge A_{{x_{2} ,n}} } \right)qB. $$

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 $Bq ( {\mathop \vee \nolimits_{i \in \Updelta } A_{\text{i}} } ).$

$$ \begin{aligned} & \Rightarrow B\left( t \right) + {\text{Max}}\{ A_{\text{i}} \left( t \right):\;{\text{i}} \in \Updelta \, \} > 1,\;{\text{for}}\;{\text{some}}\;t \in R \\ & \Rightarrow BqA_{i} \;\;{\text{for}}\;{\text{some}}\;i \in \Updelta . \\ \end{aligned} $$

(2)

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 ∈ ∆

$$ \begin{aligned} & \Rightarrow B\left( t \right) + A_{{x_{i} ,n}} \left( t \right) > 1 \\ & \Rightarrow B\left( t \right) + \mathop {\rm Max}\limits_{i \in \Updelta } \left\{ {A_{{x_{i} ,n}} \left( t \right) } \right\} > 1 \\ & \Rightarrow B\left( t \right) + \mathop \vee \limits_{i \in \Updelta } A_{{x_{i} ,n}} \left( t \right) > 1 \\ & \Rightarrow B{\text{q}}\mathop \vee \limits_{i \in \Updelta } A_{{x_{i} ,n}} \\ \end{aligned} $$

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.

$$ \begin{aligned} & \Rightarrow \lambda + {\text{B}}\left( x \right) > 1. \\ & \Rightarrow x_{\lambda } qB. \\ \end{aligned} $$

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.