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\noindent \textbf{$\lambda$- CLOSED MAPS IN INTUITIONISTIC FUZZY TOPOLOGICAL SPACES}
\noindent \textbf{P. Rajarajeswari, G. Bagyalakshmi}
\noindent \textbf{Abstract. The purpose of this paper is to introduce and study the concepts of intuitionistic fuzzy contra weakly generalized continuous map- pings in intuitionistic fuzzy topological space. Some of their properties are explored.}
\noindent \textbf{2010 AMS Classi?cation: 54A40, 03E72}
\noindent \textbf{Keywords: Intuitionistic fuzzy topology, intuitionistic fuzzy weakly generalized closed set, intuitionistic fuzzy weakly generalized open set and intuitionistic fuzzy contra weakly generalized continuous mappings. }
\noindent \textbf{Corresponding Author: G. Bagyalakshmi (g\[email protected])}
\noindent \textbf{ABSTRACT: }In this paper we introduce the concept intuitionistic fuzzy $\lambda$-open maps and intuitionistic fuzzy $\lambda$-closed maps in intuitionistic fuzzy topological space and study some of their properties.
\noindent \textbf{Keywords: }Intuitionistic fuzzy topology, intuitionistic fuzzy $\lambda$-closed maps, intuitionistic fuzzy $\lambda$-open maps.
\noindent \textbf{AMS subject classification (2000): 54A40, 03F55}
\noindent \textbf{1. INTRODUCTION}
\noindent After the introduction of fuzzy sets by L.A Zadeh [20] in 1965, there have been a number of generalizations of this fundamental concept. The concepts of intuitionistic fuzzy sets was introduced by Atanassov [1] as a generalization of fuzzy set in 1983, Coker [3] introduced the notion of intuitionistic fuzzy topology in 1997. This approach provides a wide field for investigation in the area of fuzzy topology and its application. The aim of this paper is to introduce intuitionistic fuzzy $\lambda$-open maps, intuitionistic fuzzy $\lambda$-closed maps and studied some of their properties.
\noindent \textbf{2. PRELIMINARIES}
\noindent \textbf{Definition 2.1 ([1]) }Let X be a nonempty fixed set. An intuitionistic fuzzy set (IFS) A in X is an object having the form A = \{$<$x, $\mu $${}_{A }$(x), $\upsilon $${}_{A}$ (x) $>$ : x? X\}, where the function $\mu $${}_{A}$ : X ? [0,1] and $\upsilon $${}_{A }$:X? [0,1] denotes the degree of membership $\mu $${}_{A}$(x) and the degree of non membership $\upsilon $${}_{A}$ (x) of each element x? X to the set A respectively and 0= $\mu $${}_{A}$ (x)+ $\upsilon $${}_{A}$ (x) = 1 for each x? X.
\noindent \textbf{Definition 2.2 ([1]) }Let A and B be intuitionistic fuzzy sets of the form
\noindent \textbf{A} = \{$<$x, $\mu $${}_{A }$(x), $\upsilon $${}_{A}$(x) $>$: x ? X\}, and form\textbf{ }B= \{$<$x, $\mu $${}_{B}$(x), $\upsilon $${}_{B}$ (x) $>$: x? X\}.Then\textbf{}
\noindent (a) A $\subseteq$ B if and only if $\mu $${}_{A}$(x) = $\mu $${}_{B }$(x) and $\nu $${}_{A}$(x) = $\nu $${}_{B}$(x) for all x $\in$ X
\noindent (b) A = B if and only if A $\subseteq$ B and B $\subseteq$ A
\noindent (c) A${}^{c}$ = \{$\langle$ x, $\upsilon $${}_{A}$(x), $\mu $${}_{A}$(x) $\rangle$ / x $\in$ X\}
\noindent (d) A $\cap$ B = \{$\langle$ x, $\mu $${}_{A}$(x) $\wedge$ $\mu $${}_{B}$(x), $\upsilon $${}_{A}$ (x) $\vee$ $\upsilon $${}_{B}$ (x) $\rangle$ / x $\in$ X\}
\noindent (e) A $\cup$ B = \{$\langle$ x, $\mu $${}_{A}$(x) $\vee$ $\mu $${}_{B}$(x), $\upsilon $${}_{A}$ (x) $\wedge$ $\upsilon $${}_{B}$ (x) $\rangle$ / x $\in$ X\}.
\noindent The intuitionistic fuzzy sets $\mathop{0}\limits_{\sim } $ = \{ $<$x, 0,1$>$ x $\in$ X \} and $\mathop{1}\limits_{\sim } $=\{ $<$x, 1, 0 $>$ x $\in$ X \} are respectively the empty set and whole set of X.
\noindent \textbf{Definition 2.39 ([5]) }An intuitionistic fuzzy topology (IFT) on X is a family of IFS which satisfying the following axioms.
\noindent (i) $\mathop{0}\limits_{\sim } $, $\mathop{1}\limits_{\sim } $ $\in$$\tau $
\noindent (ii) G${}_{1}$ $\cap$ G${}_{2}$ $\in$ $\tau $ for any G${}_{1}$, G${}_{2}$ $\in $$\tau $
\noindent (iii) $\cup$ G${}_{i}$ $\in$ $\tau $ for any family \{G${}_{i }$/ i $\in$ I\} $\subseteq $ $\tau $
\noindent In this case the pair (X, $\tau $) is called an intuitionistic fuzzy topological space (IFTS) and each intuitionistic fuzzy set in $\tau $ is known as an intuitionistic fuzzy open set (IFOS) in X.
\noindent The complement A of an IFOS in an IFTS (X${}_{,}$ $\tau $) is called an intuitionistic fuzzy closed set (IFCS) in (X${}_{,}$ $\tau $).
\noindent \textbf{Definition 2.4([3]) }Let\textbf{ }(X${}_{,}$ $\tau $) be an intuitionistic fuzzy topology and
\noindent A = \{$<$x, $\mu $${}_{A }$(x), $\upsilon $${}_{B}$ (x) $>$: x ? X\}, be an intuitionistic fuzzy set in X. Then the intuitionistic fuzzy interior and intuitionistic fuzzy closure are defined by
\noindent int (A) = $\cup$ \{G/ G is an intuitionistic fuzzy open set in X and G $\subseteq$ A\}
\noindent cl (A) =$\cap$ \{ K/ K in an intuitionistic fuzzy closed set in X and A$\subseteq$ K \}
\noindent \textbf{Definition 2.5 (}[11]) Let f be a mapping from an IFTS (X, $\tau $) into an IFTS (Y,$\sigma $). Then f is said to be an
\noindent (i) intuitionistic fuzzy open mapping (IF open mapping) if f(A) is an IFOS in Y for every IFOS A in X.
\noindent (ii) intuitionistic fuzzy closed mapping (IF closed mapping) if f(A) is an IFCS in Y for every IFCS A in X.
\noindent Re\textbf{mark 2.6 ([3]) }For any intuitionistic fuzzy set A in (X${}_{,}$ $\tau $), we have
\begin{enumerate}
\item cl (A${}^{C}$) = [int (A)]${}^{C}$,
\item int (A${}^{C}$) = [cl (A)]${}^{C}$,
\item A is an intuitionistic fuzzy closed set in X $\Leftrightarrow$ Cl (A) = A
\item A is an intuitionistic fuzzy open set in X $\Leftrightarrow$ int (A) = A
\end{enumerate}
\noindent \textbf{Definition 2.7 ([6]) }An intuitionistic fuzzy set A = \{$<$x, $\mu $${}_{A }$(x), $\upsilon $${}_{B}$ (x) $>$: x ? X\} in an intuitionistic fuzzy topology space (X, $\tau $) is said to be
\noindent (i) intuitionistic fuzzy semi closed if int (cl (A)) $\subseteq$ A
\noindent (ii) intuitionistic fuzzy pre closed if cl (int (A))$\subseteq$ A
\noindent \textbf{Definition 2.8} ([5]) Let X and Y are nonempty sets and f: X$?$Y is a function
\noindent (a) If B = \{$<$ y, $\mu $${}_{B }$(y), $\upsilon $${}_{B}$ (y) $>$ : y$\in$Y\} is an intuitionistic fuzzy set in Y, then the pre image of B under f denoted by f${}^{-1}$(B), is defined by
\noindent f${}^{-1 }$(B) ==\{$<$x,f ${}^{-1}$($\mu $ ${}_{B}$(x)),f ${}^{-1}$($\upsilon $${}_{B}$(x)) $>$ : x $\in$ X\}
\noindent (b) If A= \{$<$\{x, $\mu $ ${}_{A}$ (x), $\upsilon $${}_{B}$ (x),)$>$ / x$\in$ X\} is an intuitionistic fuzzy set in X, the image of A under f denoted by f(A) is the intuitionistic fuzzy set in Y defined by
\noindent f(A) = \{$<$y, f ($\mu $${}_{A }$(y)), f ($\upsilon $${}_{A}$(y)) $>$ : y $\in$ Y\} where f ($\upsilon $${}_{A}$) = 1-f(1-($\upsilon $${}_{A}$))\textbf{}
\noindent \textbf{Definition 2.9 (}[9])\textbf{ }An intuitionistic fuzzy set A of an intuitionistic topology space (X, \tau ) is called an
\noindent (i) intuitionistic fuzzy $\lambda$-closed set (IF $\lambda$-CS) if A $\supseteq $ cl(U) whenever A$\supseteq $U and U is intuitionistic fuzzy open set in X.
\noindent (ii)\textbf{ }intuitionistic fuzzy $\lambda$-open set (IF $\lambda$-OS) if the complement $A^{c} $ is an intuitionistic fuzzy $\lambda$-closed set A.
\noindent \textbf{Definition 2.10: }An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X,$\tau$) called
\noindent (i) intuitionistic fuzzy generalized closed set [15] (intuitionistic fuzzy g -- closed) if cl(A) $\subseteq$ U whenever A $\subseteq$ U and U is intuitionistic fuzzy semi open
\noindent (ii) intuitionistic fuzzy g -- open set [14], if the complement of an intuitionistic fuzzy g -- closed set is called intuitionistic fuzzy g - open set.
\noindent (iii) intuitionistic fuzzy semi open (resp. intuitionistic fuzzy semi closed) [6] if there exists an intuitionistic fuzzy open (resp. intuitionistic fuzzy closed) such that U$\subseteq$A $\subseteq$ Cl(U) (resp. int(U) $\subseteq$ A $\subseteq$ U).
\noindent \textbf{Definition 2.11:} An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space\textbf{ }(X,\includegraphics*[width=0.14in, height=0.20in, keepaspectratio=false]{image208}) is called
\noindent (i) an intuitionistic fuzzy w-closed \textbf{[14]} if cl(A) $\subseteq$ O whenever A$\subseteq$O and O is intuitionistic fuzzy semi open. (X,$\tau $)
\noindent (ii) an intuitionistic fuzzy generalized $\alpha $-closed set [8] (IFG$\alpha $CS if$\ \alpha $cl(A) $\subseteq$ O whenever A$\subseteq$O and O is IF$\alpha $OS in(X,$\tau $)
\noindent (iii) an intuitionistic fuzzy $\alpha $-generalized closed set [12] (IF$\alpha $GCS if$\ \alpha $cl(A) $\subseteq$ O whenever A$\subseteq$O and O is IFOS in(X,$\tau $)
\noindent (iv) an intuitionistic fuzzy regular closed set [4 ] (IFRCS in short) if A = cl(int(A)),
\noindent (v) an intuitionistic fuzzy regular open set [4 ](IFROS in short) if A = int(cl(A)),
\noindent \textbf{Definition 2.12 (}[10]) A mapping f: (X,$\tau $) ? (Y, $\sigma $) is said to be intuitionistic fuzzy \textbf{$\lambda$} --continuous if the inverse image of every intuitionistic fuzzy closed set of Y is intuitionistic fuzzy \textbf{$\lambda$} -closed in X.
\noindent \textbf{Definition 2.13 ([11]) }A topological space (X, $\tau $) is called intuitionistic fuzzy \textbf{$\lambda$ - }\textit{T${}_{1/2}$${}_{ }$}space
\noindent (IF \textbf{$\lambda$ -} \textit{T${}_{1/2}$}\textbf{ s}pace in short) if every intuitionistic fuzzy \textbf{$\lambda$}-closed set is intuitionistic fuzzy closed in X.
\noindent \textbf{Definition 2.14: }A mapping \textit{f}: (X,$\tau $) ? (Y, $\sigma $) is said to be
\noindent (i) an intuitionistic fuzzy w-closed [15 ] if image of every intuitionistic fuzzy closed set of X is intuitionistic fuzzy w-closed set in Y
\noindent (ii) an intuitionistic fuzzy regular closed [17 ] if image of every intuitionistic fuzzy closed set of X is intuitionistic fuzzy regular closed set in Y.
\noindent (iii) an intuitionistic fuzzy generalized $\alpha $-closed [8] if image of every intuitionistic fuzzy closed set of X is intuitionistic fuzzy generalized $\alpha $-closed set in Y
\noindent (iv) an intuitionistic fuzzy $\alpha $-generalized closed [12] if image of every intuitionistic fuzzy closed set of X is intuitionistic fuzzy $\alpha $-generalized closed set in Y
\noindent \textbf{Definition 2.15 ([10]) }Let \textit{A }be an IFS in an IFTS (\textit{X}, $\tau $). Then the Intuitionistic fuzzy \textbf{$\lambda$-}interior and intuitionistic fuzzy \textbf{$\lambda$-c}losure of A are defined as follows.
\noindent $\lambda$-int(\textit{A}) = $\cup $\{\textit{G }\textbar \textit{G }is an IF$\lambda$-OS in \textit{X }and \textit{G }$\subseteq $ \textit{A}\}
\noindent $\lambda$-cl(\textit{A}) = n\{\textit{K }\textbar \textit{K }is an IF$\lambda$-CS in \textit{X }and \textit{A}$\subseteq $\textit{ K }\}
\noindent \textbf{3. INTUITIONISTIC FUZZY $\lambda$-CLOSED MAPPINGS}
\noindent \textbf{Definition 3.1:} A mapping f: (X, \tau ) ? (Y, $\sigma $) is said to be intuitionistic fuzzy \textbf{$\lambda$-}closed map (IF \textbf{$\lambda$}-- closed map) if f(V) is $\lambda$-closed in (Y, $\sigma $) for every closed set V in (X, $\tau $).
\noindent \textbf{Theorem 3.2: }Every IF closed map is an IF \textbf{$\lambda$-}closed map but not conversely.
\noindent \textbf{Proof: }Let \textit{f}: \textit{X }? \textit{Y }be an IF closed map. Let \textit{A }be an IFCS in \textit{X}. Then \textit{f}(\textit{A}) is an IFCS in \textit{Y}. Since every IFCS is an IF \textbf{$\lambda$}--CS, \textit{f}(\textit{A}) is an IF\textbf{$\lambda$}--CS in \textit{Y}[8].Hence \textit{f }is an IF\textbf{$\lambda$-}closed map.
\noindent \textbf{Remark 3.3: }The converse above theorem need not be true as seen from the following example.
\noindent \textbf{Example 3.4: }Let X = \{ a, b\} and Y = \{u, v\}\textbf{ }intuitionistic fuzzy sets U and V are defined as follows; U=\{$<$a,0.5,0.5.$>$,$<$b,0.2,0.8$>$\} V=\{$<$u,0.5,0.5$>$, $<$v,0.3,0.6$>$\}
\noindent Let $\tau $ =\{0,1,U\} and $\sigma $ =\{0,1,V\} be Intuitionistic fuzzy topologies on X and Y respectively. Define a map f: (X, $\tau $) ? (Y,$\ \sigma $ ) by f(a) =u and f(b) = v.
\noindent Then f (U)=f(\{$<$a,0.5,0.5$>$, $<$b,0.8,0.2$>$\}) = \{$<$u,0.5,0.5$>$, $<$v,0.8,0.2 $>$\} is $\lambda$ -closed set in Y But \{$<$u,0.5,0.5$>$, $<$v,0.8,0.2$>$\} is not closed set in Y. Hence f is intuitionistic fuzz $\lambda$ - closed mapping in Y but not intuitionistic fuzzy closed in Y
\noindent \textbf{Theorem 3.6: }Let \textit{f }: \textit{X }? \textit{Y }be an IF$\lambda$-closed map where \textit{Y }is an IF$\lambda$-T${}_{1/2}$ space, then \textit{f }is an IF closed map if every IF$\lambda$-CS is an IFCS in \textit{Y}.
\noindent \textbf{Proof: }Let \textit{f }be an IF$\lambda$-closed map. Then for every IFCS \textit{A }in \textit{X}, \textit{f}(\textit{A}) is an IF$\lambda$-CS in Y. Since Y is an IF$\lambda$-T${}_{1/2}$ space, f(A) is an IF$\lambda$-CS in Y and by hypothesis f(A) is an IFCS in Y. Hence f is an IF closed map.
\noindent \textbf{Theorem 3.7:} Every IF pre closed map is IF $\lambda$ -closed map.
\noindent \textbf{Proof: }Let \textit{f}: \textit{X }? \textit{Y }be an IF pre closed map. Let \textit{A }be an IFCS in \textit{X}. By assumption \textit{f}(\textit{A}) is an IF pre closed set in \textit{Y}. Since every IF pre closed set is an IF \textbf{$\lambda$}--CS [6] \textit{f}(\textit{A}) is an IF\textbf{$\lambda$}--CS in \textit{Y}. Hence \textit{f }is an IF$\lambda$-closed map.\textbf{}
\noindent \textbf{Remark 3.8:} The converse above theorem need not be true as seen from the following example:
\noindent \textbf{Example 3.9}: Let X = \{a,b\}\textbf{ }and Y=\{u,v\} and intuitionistic fuzzy sets U and V are defined as follows; U= \{$<$ a, 0.5,0.5$>$, $<$ b, 0.2,0.8 $>$\} and V=\{$<$u,0.5,0.5$>$, $<$v, 0.3, 0.7$>$\}
\noindent Let \tau =\{0,1,U\} and $\sigma $=\{0,1,V\} be intuitionistic fuzzy topologies on X and Y respectively. Define a map f:(X, \tau ) ? (Y,$\ \sigma $ ) by f(a) =u and f(b) = v.
\noindent Then f (U)=f(\{$<$a,0.5,0.5$>$, $<$b,0.8,0.2$>$\}) = \{$<$u,0.5,0.5$>$, $<$v,0.8,0.2 $>$\} is IF $\lambda$-closed set in Y but not IF preclosed set in Y. Hence f is intuitionistic fuzzy $\lambda$ -closed map but not intuitionistic fuzzy pre closed map.
\noindent \textbf{Remark 3.10}: IF $\lambda$ -closed map and IF w-closed map are independent to each other as seen from the following example.
\noindent \textbf{Example 3.11}: Let X = \{a,b\} and Y=\{u,v\} and intuitionistic fuzzy sets U and V are defined as follows U = \{$<$a, 0.5, 0.5$>$, $<$b, 0.5,0.4$>$\} and V = \{$<$a, 0.5, 0.5$>$, $<$b, 0.5, 0.2$>$\}.
\noindent Let \tau =\{0,1,U\} and $\sigma $ =\{0,1,V\} be intuitionistic fuzzy topologies on X and Y respectively. Define a map f:(X, \tau ) ? (Y,$\ \sigma $ ) by f(a) =u and f(b) = v.
\noindent Then f(U) =f($<$a, 0.5, 0.5$>$, $<$b, 0.4,0.5$>$\} = ($<$u, 0.5, 0.5$>$, $<$v, 0.4,0.5$>$\} is intuitionistic fuzzy IF $\lambda$-closed set but not IF w-closed set Hence f is intuitionistic fuzzy $\lambda$-closed mapping but not intuitionistic fuzzy w-closed mappings.
\noindent \textbf{Example 3.12}: Let X = \{a,b\} and Y=\{u,v\} and intuitionistic fuzzy sets U and V are defined as follows. U=\{$<$a, 0.5, 0.5$>$, $<$b, 0.5,0.5 $>$\} and V = \{$<$u, 0.5, 0.5$>$, $<$v, 0.4, 0.6 $>$\}.
\noindent Let \tau =\{0,1,U\} and $\sigma $ =\{0,1,V\} be intuitionistic fuzzy topologies on X and Y respectively. Define a map f:(X, \tau ) ? (Y,$\ \sigma $ ) by f(a) =u and f(b) = v .
\noindent f(U)=f\{( $<$ a,0.5,0.5$>$,$<$b, 0.5, 0.5$>$)\}= \{$<$u,0.5,0.5$>$,$<$v, 0.5, 0.5$>$ \} is IF w-closed set not IF $\lambda$-closed set. Hence f is intuitionistic fuzzy w- closed mapping but not intuitionistic fuzzy $\lambda$ - closed mappings.
\noindent \textbf{Remark 3.13:} Intuitionistic fuzzy g-closed mappings and Intuitionistic fuzzy $\lambda$ - closed mappings are independent as seen from the following examples.
\noindent \textbf{Example 3.14: }Let X = \{a, b\}, Y=\{u, v\} and intuitionistic fuzzy sets U and V are defined as follows. U=\{$<$a,0.5,0.5$>$,$<$b,0.6,0.3\}, V=\{$<$a,0.5,0.5$>$,$<$b,0.2,0.6\}.
\noindent Let $\tau $ = \{$\mathop{0}\limits_{\sim } $,$\mathop{1}\limits_{\sim } $, U \} and $\sigma $ =\{$\mathop{0}\limits_{\sim } $ $\mathop{1}\limits_{\sim } $, V\} be intuitionistic fuzzy topologies on X and Y respectively.Define map: f: (X,$\tau $) ? (Y,$\sigma $) by f(a)=u and f(b)=v then f\{($<$a,0.5,0.5$>$,$<$b,0.3,0.6$>$\}= \{$<$a,0.5,0.5$>$,$<$b,0.3,0.6 \} is intuitionistic fuzzy g-closed set but not intuitionistic fuzzy --$\lambda$ closed set. Hence f is intuitionistic fuzzy g-closed mapping and not intuitionistic fuzzy $\lambda$- mapping.
\noindent \textbf{Example 3.15 : }Let X X= \{a,b \} and Y=\{u, v\} and intuitionistic fuzzy sets U and V are defined as follows U=\{$<$a,0.5,0.5$>$,$<$b,0.5, 0.2 $>$\} and V=\{$<$a,0.5,0.5$>$, $<$b,0.5,0.4$>$\}. Let \includegraphics*[width=0.14in, height=0.20in, keepaspectratio=false]{image223}= \{ $\mathop{0}\limits_{\sim } $ $\mathop{1}\limits_{\sim } $ U\} and $\sigma $ = \{ $\mathop{0}\limits_{\sim } $ $\mathop{1}\limits_{\sim } $ ,V\} be intuitionistic fuzzy topologies on X and Y respectively. Define the map f : X (X,$\tau $) $?$ (Y,$\sigma $) by f(a)=u and f(b)=v. Then f\{($<$a,0.5,0.5$>$, $<$b,0.2, 0.5 $>$\} = \{$<$a,0.5,0.5$>$, $<$b,0.2, 0.5 $>$\} is intuitionistic fuzzy $\lambda$-closed set but not intuitionistic fuzzy g closed set. Hence f is intuitionistic fuzzy $\lambda$- mappings and not intuitionistic fuzzy g-closed mappings.
\noindent \textbf{Remark 3.16: }The concept of intuitionistic fuzzy $\lambda$- closed mappings and intuitionistic fuzzy semi closed mappings are independent as seen from the following examples.
\noindent \textbf{Example 3.17: }Let X=\{a,b\}, Y=\{u, v\} and intuitionistic fuzzy sets U and V are defined as follows: U=\{$<$a, 0. 5, 0 .5 $>$, $<$b, 0.6, 0 .4$>$\},V= \{$<$u, 0.8,0.2 $>$, $<$v, 0.1, 0 .9 $>$\}. Let \includegraphics*[width=0.14in, height=0.20in, keepaspectratio=false]{image227} = \{$\mathop{0}\limits_{\sim } $, $\mathop{1}\limits_{\sim } $,U\} and $\sigma $ =\{ $\mathop{0}\limits_{\sim } $, $\mathop{1}\limits_{\sim } $ ,V\} be intuitionistic fuzzy topologies on X and Y respectively. Define the map f: (X,$\tau $) $?$ (Y,$\sigma $) by f(a)=u and f(b)=v then the mapping f($<$a, 0. 5, 0 .5$>$, $<$b, 0.4, 0 .6$>$)=\{$<$u, 0. 5, 0 .5 $>$, $<$v, 0.4, 0 .6$>$\} is intuitionistic fuzzy semi closed set and not intuitionistic fuzzy $\lambda$ - closed set . Hence f is intuitionistic fuzzy semi closed mapping but not intuitionistic fuzzy $\lambda$ - closed.
\noindent \textbf{Example 3.18: }Let X=\{a,b\}, Y=\{u,v \} and intuitionistic fuzzy sets U and V are defined as follows: U= \{$<$a, 0.5, 0.5$>$, $<$b, 0.5, 0.2$>$\}, V= \{a, 0 .5, 0.5$>$, $<$b, 0.5, 0 .4$>$\}.
\noindent Let\includegraphics*[width=0.18in, height=0.16in, keepaspectratio=false]{image231}=\{ $\mathop{0}\limits_{\sim } $ $\mathop{1}\limits_{\sim } $, U\} and $\sigma $=\{ $\mathop{0}\limits_{\sim } $ $\mathop{1}\limits_{\sim } $ ,V\}be intuitionistic fuzzy topologies on \textit{X} and Y respectively. Define the map f : (X,$\tau $) $?$ (Y,$\sigma $) by f(a)=x and f(b)=y then f(U) =f\{($<$a, 0.5, 0.5$>$, $<$b, 0.2, 0.5 $>$)\} =\{($<$u, 0.5, 0.5$>$, $<$v, 0.2, 0.5 $>$)\} is intuitionistic fuzzy $\lambda$- closed set but not intuitionistic fuzzy semi closed set.
\noindent Hence f is intuitionistic fuzzy $\lambda$-closed mappings but not intuitionistic fuzzy semi closed mapping.
\noindent \textbf{Remark 3.19: }The concept of intuitionistic fuzzy $\lambda$- closed mappings and intuitionistic fuzzy semi pre closed mappings are independent as seen from the following examples.
\noindent \textbf{Example 3.20: }Let X=\{a,b,\}, Y=\{u, v\} and intuitionistic fuzzy sets U and V are defined as follows: U= \{$<$a, 0.5, 0.5$>$, $<$b, 0.3, 0.5 $>$\}, V= \{u, 0 .5, 0.5$>$, $<$v, 0.5, 0 .3 $>$\}.
\noindent Let =\{$\mathop{0}\limits_{\sim } $, $\mathop{1}\limits_{\sim } $ ,U \} and $\sigma $ =\{$\mathop{0}\limits_{\sim } $ $\mathop{1}\limits_{\sim } $ ,V\}be intuitionistic fuzzy topologies on X and Y respectively. Define the map f: (X,$\tau $) $?$ (Y,$\sigma $) by f(a)=u and f(b)=v then f($<$a, 0.5, 0.5$>$, $<$b, 0.5, 0.3 $>$\} =$<$u, 0.5, 0.5$>$, $<$v, 0.5, 0.3 $>$\} is intuitionistic fuzzy $\lambda$-closed set, but not intuitionistic fuzzy semi pre closed set. Then f is intuitionistic fuzzy $\lambda$-closed mapping, but not intuitionistic fuzzy semi pre closed mapping.
\noindent \textbf{Example 3.21: }Let X=\{a, b\}, Y=\{u,v\} and intuitionistic fuzzy sets U and V are defined as follows: U= \{$<$a, 0. 5, 0 .5 $>$, $<$b, 0. 4, 0 .6$>$\} and V = \{$<$u, 0.2,0.8 $>$, $<$v, 0 .1, 0 .9$>$\}.
\noindent Let \includegraphics*[width=0.14in, height=0.20in, keepaspectratio=false]{image238}=\{$\mathop{0}\limits_{\sim } $, $\mathop{1}\limits_{\sim } $, U\} and $\sigma $=\{$\mathop{0}\limits_{\sim } $ $\mathop{1}\limits_{\sim } $,V\} be intuitionistic fuzzy topologies on X and Y respectively. Define the map f: (X,$\tau $) $?$ (Y,$\sigma $) by f(a)=u and f (b)=v.
\noindent Then f (\{$<$a, 0. 5, 0 .5 $>$, $<$b, 0. 4, 0 .6$>$\})=\{$<$u, 0. 5, 0 .5 $>$, $<$v, 0. 4, 0 .6$>$\} is intuitionistic fuzzy semi pre closed mapping but not intuitionistic fuzzy $\lambda$-closed mapping.
\noindent \textbf{Remark 3.22}: IF G$\alpha $-closed mapping and IF $\lambda $-closed mappings are independent to each other.
\noindent \textbf{Example 3.23:} Let X = \{a,b\} and Y=\{u,v \}and intuitionistic fuzzy sets U and V are defined as follows; U= \{$<$ a, 0.4 ,0.6 $>$, $<$ b, 0.3,0.7 $>$ \},V=\{$<$ a,0.2,0.8 $>$,$<$b, 0.3, 0.7 $>$ \}
\noindent Let \includegraphics*[width=0.14in, height=0.20in, keepaspectratio=false]{image242}=\{0,1,U\} and $\sigma $ =\{0,1,V\} be intuitionistic fuzzy topologies on X and Y respectively. Define a map f: (X,$\tau $) $?$ (Y,$\sigma $) by f(a) =u and f(b) = v
\noindent Then f(\{$<$a,0.6,0.4$>$, $<$b,0.7,0.3$>$\}) = \{\{$<$a,0.6,0.4$>$, $<$b,0.7,0.3$>$\}) is IF G$\alpha $-closed set but not IF-closed set. Hence f is intuitionistic fuzzy $\lambda$-mapping but not intuitionistic fuzzy IF G$\alpha $-closed mapping.
\noindent \textbf{Example 3.24 :} Let X = \{a,b\} and Y=\{u,v \}and intuitionistic fuzzy sets U and V are defined as follows; U= \{$<$ a, 0.1,0.9$>$, $<$ b, 0.3,0.7$>$\}.V=\{$<$ a,0.8,0.2$>$, $<$b, 08, 0.1$>$\}
\noindent Let \includegraphics*[width=0.14in, height=0.20in, keepaspectratio=false]{image246}=\{0,1,U\} and $\sigma $ =\{0,1,V\} be intuitionistic fuzzy topologies on X and Y respectively. Define a map f: (X,$\tau $) $?$ (Y,$\sigma $) by f(a) =u and f(b) = v
\noindent Then f(\{$<$a,0.9,0.1 $>$, $<$b,0.7,0.3 $>$\}) = \{$<$a,0.9,0.1 $>$, $<$b,0.7,0.3 $>$\})${\rm \ IF\ }{\rm -}{\rm closed\ set}$ in Y but is not $IF{\rm G}\alpha {\rm closed\ set}$. Hence f is intuitionistic fuzzy $\lambda$- mapping but not intuitionistic fuzzy$\ IF{\rm G}\alpha {\rm closed}$ mapping.
\noindent \textbf{Remark 3.25}: IF${\rm G}\alpha $-closed mapping ${\rm and\ }$IF $\lambda$-closed mapping are independent to each other as seen from the following example:
\noindent \textbf{Example 3.26: }Let X = \{a,b\} and Y=\{u,v \}and intuitionistic fuzzy sets U and V are defined as follows; U= \{$<$ a, 0.6,0.4 $>$, $<$ b, 0.7,0.2 $>$ \}.V=\{$<$u, 0.2, 0.6 $>$,$<$b, 0.2, 0.7 $>$ \}
\noindent Let \includegraphics*[width=0.14in, height=0.20in, keepaspectratio=false]{image250}=\{0,1,U\} and $\sigma $ =\{0,1,V\} be intuitionistic fuzzy topologies on X and Y respectively. Define a map f: (X,$\tau $) $?$ (Y,$\sigma $) ? (Y,$\ \sigma $ ) by f(a) =u and f(b) = v
\noindent Then f (U)=f(\{$<$ a, 0.6,0.4$>$,$<$b, 0.7,0.2 $>$\}=\{$<$a,0.6,0.4$>$,$<$b,0.7,0.2 $>$\}in${\rm \ }\alpha {\rm G}{\rm \ closed\ set\ }$Y.but not $\lambda$ -closed set in Y . Hence f is intuitionistic fuzzy IF$\alpha {\rm G}$-closed mapping, but not IF$\lambda$-closed mapping.
\noindent \textbf{Example 3.27: }Let X = \{a,b\} and Y=\{u,v \}and intuitionistic fuzzy sets U and V are defined as follows; U= \{$<$ a, 0.2,0.8 $>$, $<$ b, 0.3,0.7 $>$ \}.V=\{$<$u, 0.6, 0.4 $>$,$<$b, 0.7, 0.3 $>$ \}
\noindent Let \includegraphics*[width=0.14in, height=0.20in, keepaspectratio=false]{image255}=\{0,1,U\} and $\sigma $ =\{0,1,V\} be intuitionistic fuzzy topologies on X and Y respectively. Define a map f: (X,$\tau $) $?$ (Y,$\sigma $) by f(a) =u and f(b) = v
\noindent Then f (U)=f(\{$<$ a, 0.8,0.2$>$,$<$b, 0.7,0.3 $>$\}=\{$<$a,0.8,0.2$>$,$<$b,0.7,0.3 $>$\}is${\rm \ }$
\noindent ${\rm \ }{\rm -}{\rm closed\ set\ }$Y, but not $\alpha {\rm G}$ -closed set in Y . Hence f is intuitionistic fuzzy $\lambda$-closed mapping but not intuitionistic fuzzy $\alpha {\rm G}{\rm \ closed\ mappings}$ .
\noindent \textbf{Remark 3.28}: From above examples and remarks we get following diagram of implications.
\noindent
\noindent
\noindent
\noindent \textbf{}
\noindent \textbf{}
\noindent \textbf{}
\noindent
\noindent
\noindent
\noindent \textbf{}
\noindent \textbf{}
\noindent \textbf{}
\noindent \textbf{}
\noindent \eject
\noindent In this diagram A B means that A implies B.
\noindent
\noindent A B means that B does not implies A\textit{.}
\noindent
\noindent A B means that A and B are independent to each other.
\noindent \textbf{Theorem 3.29: }Let \textit{f}: \textit{X }? \textit{Y }be a mapping. Then the following are equivalent if \textit{Y }is an
\noindent IF$\lambda$- T${}_{1/2 }$space.
\noindent (i) \textit{f }is an IF$\lambda$- closed map
\noindent (ii) $\lambda$- cl (\textit{f}(\textit{A})) $\subseteq $ \textit{f}(cl (\textit{A})) for each IFS \textit{A }of \textit{X}.
\noindent \textbf{Proof: }(i) $\Rightarrow $ (ii)\textit{ }Let \textit{A }be an IFS in \textit{X}. Then cl(\textit{A}) is an IFCS in \textit{X}. (i) implies that \textit{f}(cl(\textit{A})) is an IF$\lambda$-CS in \textit{Y}. Since \textit{Y }is an IF$\lambda$-T${}_{1/2 }$space, \textit{f}(cl(\textit{A})) is an IFCS in \textit{Y}. Therefore $\lambda$-cl(\textit{f}(cl (\textit{A}))) = \textit{f}(cl(\textit{A})). Now$\lambda$- cl (\textit{f}(\textit{A})) $\subseteq $$\lambda$-cl (\textit{f}(cl (\textit{A}))) = \textit{f}(cl (\textit{A})).
\noindent Hence $\lambda$-cl(\textit{f}(\textit{A}))$\ \subseteq $ \textit{f}(\textit{cl}(\textit{A})) for each IFS \textit{A }of \textit{X}.
\noindent (ii)$\Rightarrow $ (i)\textit{ }Let \textit{A }be any IFCS in \textit{X}. Then cl(\textit{A}) = \textit{A}. (ii)\textit{ }implies that
\noindent $\lambda$-cl(\textit{f}(\textit{A})) $\subseteq $ \textit{f}(cl(\textit{A})) = \textit{f}(\textit{A}).But \textit{f}(\textit{A}) $\subseteq $$\lambda$- cl\textit{(f}(\textit{A})). Therefore$\lambda$- cl(\textit{f}(\textit{A})) = \textit{f}(\textit{A}). This implies \textit{f}(\textit{A}) is an IF$\lambda$-CS in\textit{Y}. Since every IF$\lambda$-CS is an IFCS, \textit{f}(\textit{A}) is an IF$\lambda$- CS in \textit{Y}. Hence \textit{f }is an IF$\lambda$- closed map.
\noindent \textbf{Theorem 3.30: }Let \textit{f }: \textit{X }? \textit{Y }be a bijection. Then the following are equivalent if \textit{Y }is an
\noindent IF$\lambda$-T${}_{1/2}$ space
\noindent (i) \textit{f }is an IF$\lambda$-closed map
\noindent (ii) $\lambda$- cl(\textit{f}(\textit{A})) $\subseteq $ \textit{f}(cl(\textit{A})) for each IFS \textit{A }of \textit{X}
\noindent (iii) \textit{f }${}^{-1}$($\lambda$-cl(\textit{B})) $\subseteq $ cl(\textit{f ${}^{-1}$}(\textit{B})) for every IFS \textit{B }of \textit{Y}.
\noindent \textbf{Proof: }(i) $\Leftrightarrow $ (ii) is obvious from Theorem 3.15.
\noindent (ii) $\Rightarrow $ (iii\textit{) }Let \textit{B }be an IFS in \textit{Y}. Then \textit{f }${}^{-1}$(\textit{B}) is an IFS in \textit{X}. Since \textit{f }is onto,
\noindent $\lambda$-cl(\textit{B}) = $\lambda$-cl(\textit{f}(\textit{f${}^{-1}$} (\textit{B}))) and (ii) implies $\lambda$-cl(\textit{f}(\textit{f ${}^{-1}$}(\textit{B}))) $\subseteq $ \textit{f}(cl(\textit{f ${}^{-1}$}(\textit{B}))). Therefore
\noindent $\lambda$-cl(\textit{B}) $\subseteq $ \textit{f}(cl(\textit{f ${}^{-1}$}(\textit{B}))). Now \textit{f }${}^{-1}$($\lambda$-cl(B)) $\subseteq $ \textit{f }${}^{-1}$(\textit{f}(cl(\textit{f }${}^{-1}$(\textit{B}))). Since \textit{f }is one to one,
\noindent \textit{f }${}^{-1}$($\lambda$-cl(\textit{B})) $\subseteq $ cl(\textit{f }${}^{-1}$(\textit{B})).
\noindent (iii) $\Rightarrow $ (ii)\textit{ }Let \textit{A }be any IFS of \textit{X}. Then \textit{f}(\textit{A}) is an IFS of \textit{Y}. Since \textit{f }is one to one,
\noindent (iii) implies that \textit{f ${}^{-1}$}($\lambda$-cl(\textit{f}(\textit{A})) $\subseteq $ cl(\textit{f ${}^{-1}$}(\textit{A})) = cl(\textit{A}). Therefore \textit{f }(\textit{f ${}^{-1}$}($\lambda$-cl(\textit{f}(\textit{A})))) $\subseteq $\textit{f}(cl(\textit{A})). Since \textit{f }is onto .$\lambda$-cl(\textit{f}(\textit{A})) = \textit{f}(\textit{f ${}^{-1}$}($\lambda$-cl(\textit{f}(\textit{A})))) $\subseteq $ \textit{f}(cl(\textit{A})).
\noindent
\noindent \textbf{Theorem 3.31: }Let \textit{f }: \textit{X }? \textit{Y }be an IF$\lambda$-closed map. Then for every IFS \textit{A }of \textit{X}, \textit{f}(cl(\textit{A})) is an
\noindent IF$\lambda$-CS in \textit{Y}.
\noindent \textbf{Proof: }Let \textit{A }be any IFS in \textit{X}. Then cl(\textit{A}) is an IFCS in \textit{X}. By hypothesis, \textit{f}(cl(\textit{A})) is an
\noindent IF$\lambda$-CS in \textit{X}.
\noindent \textbf{Theorem 3.32: }Let \textit{f }: \textit{X }? \textit{Y }be an IF$\lambda$-closed map where \textit{Y }is an IF$\lambda$-T${}_{1/2}$ space. Then \textit{f }is a
\noindent IF regular closed map if every IF$\lambda$-CS is an IFRCS in \textit{Y}.
\noindent \textbf{Proof: }Let \textit{A }be an IFRCS in \textit{X}. since every IFRCS is an IFCS[5], \textit{A }is an IFCS in \textit{X}. By hypothesis \textit{f}(\textit{A}) is an IF$\lambda$-CS in \textit{Y}. Since \textit{Y }is an IF$\lambda$-T${}_{1/2 }$space, \textit{f}(\textit{A}) is an IF$\lambda$-CS in \textit{Y }and hence is an IFCS in \textit{Y}, by hypothesis. This implies that \textit{f}(\textit{A}) is an IF regular closed map.
\noindent \textbf{Theorem 3.33 : }If every IFS is an IFCS, then an IF$\lambda$-closed mapping is an IF $\lambda$- continuous mapping.
\noindent \textbf{Proof: }Let \textit{A }be an IFCS in \textit{Y}. Then \textit{f }${}^{-1}$(\textit{A}) is an IFS in \textit{X}. Therefore \textit{f}${}^{--1}$ (\textit{A}) is an IFCS in \textit{X}. Since every IFCS is an IF$\lambda$-CS, \textit{f }${}^{-1}$(\textit{A}) is an IF$\lambda$-CS in \textit{X}. This implies that \textit{f }is an IF$\lambda$-continuous mapping.
\noindent \textbf{Theorem 3.34 : }A mapping \textit{f }: \textit{X }? \textit{Y }is an IF$\lambda$-closed mapping if and only if for every IFS \textit{B }of \textit{Y }and for every IFOS \textit{U }containing \textit{f }${}^{-1}$(\textit{B}), there is an IF$\lambda$-OS \textit{A }of \textit{Y }such that \textit{B }$\subseteq $\textit{A }and
\noindent \textit{f }${}^{-1}$ (\textit{A}) $\subseteq $ \textit{U}.
\noindent \textbf{Proof}: \textbf{Necessity}\textit{: }Let \textit{B }be any IFS in \textit{Y}. Let \textit{U }be an IFOS in \textit{X }such that \textit{f ${}^{-1}$}(\textit{B}) $\subseteq $\textit{U}, then \textit{U${}^{c}$${}^{ }$}is an IFCS in \textit{X}. By hypothesis \textit{f}(\textit{U${}^{c}$}) is an IF$\lambda$-CS in \textit{Y}. Let A = (\textit{f}(\textit{U${}^{c}$}))${}^{c}$, then \textit{A }is\textit{${}^{ }$}an IF$\lambda$-OS in \textit{Y }and B $\subseteq $A. Now \textit{f ${}^{-1}$}(\textit{A}) = \textit{f }${}^{-1}$(\textit{f(U${}^{c}$}))${}^{c}$\textit{ }= (\textit{f }${}^{-1}$(\textit{f}(\textit{U${}^{c}$})))\textit{${}^{c}$ }$\subseteq $ \textit{U}.\textit{}
\noindent \textbf{Sufficiency}\textit{: }Let \textit{A }be any IFCS in \textit{X}, then \textit{A${}^{c}$ }is an IFOS in \textit{X }and \textit{f ${}^{-1}$}(\textit{f}(\textit{A${}^{c}$}))${}^{c}$$\subseteq $\textit{ A${}^{c}$}. By hypothesis there exists an IF$\lambda$-OS \textit{B }in \textit{Y }such that \textit{f}(\textit{A${}^{c}$}) $\subseteq $ \textit{B }and \textit{f ${}^{-1}$}(\textit{B}) $\subseteq $ \textit{A${}^{c}$}. Therefore \textit{A }$\subseteq $ (\textit{f }${}^{-1}$(\textit{B}))${}^{c}$. Hence \textit{B ${}^{c}$}$\subseteq $ \textit{f}(\textit{A}) $\subseteq $\textit{f}(\textit{f ${}^{-1}$}(\textit{B})) ${}^{c}$\textit{ }$\subseteq $ \textit{B ${}^{c}$}. This implies that \textit{f}(\textit{A}) = \textit{B${}^{c}$.}Since \textit{B${}^{ c}$ }is an IF$\lambda$-CS in \textit{Y}, \textit{f}(\textit{A}) is an IF$\lambda$-CS in \textit{Y}. Hence \textit{f }is an IF$\lambda$-closed mapping.
\noindent \textbf{Theorem 3.35: }If \textit{f}: \textit{X }? \textit{Y }is an IF closed map and \textit{g}: \textit{Y}? \textit{Z }is an IF$\lambda$-closed map, then \textit{g }$o $ \textit{f }is an IF$\lambda$- closed map.
\noindent \textbf{Proof: }Let \textit{A }be an IFCS in \textit{X}, then \textit{f}(\textit{A}) is an IFCS in \textit{Y}, Since \textit{f }is an IF closed map. Since \textit{g }is an IF$\lambda$-closed map, \textit{g}(\textit{f}(\textit{A})) is an IF$\lambda$-CS in \textit{Z}. Therefore \textit{g }$o $ \textit{f }is an IF$\lambda$-closed map.
\noindent \textbf{Theorem 3.36: }Let \textit{f}: \textit{X }? \textit{Y }be a bijective map where \textit{Y }is an IF$\lambda$-T${}_{1/2}$ space. Then the following are equivalent.
\noindent (i) \textit{f }is an IF$\lambda$-closed map.
\noindent (ii) \textit{f }(\textit{B}) is an IF$\lambda$-OS in \textit{Y }for every IFOS \textit{B }in \textit{X}.
\noindent \textbf{Proof: }(i) $\Leftrightarrow $ (ii) is obvious.
\noindent \textbf{Definition 3.37:} A mapping f: (X,$\tau $) $?$ (Y,$\sigma $) is said to be intuitionistic fuzzy \textbf{$\lambda$}--open map (IF \textbf{$\lambda$}--open map) if f(V) is $\lambda$-open set in (Y,${\rm \ }\sigma {\rm ),for\ every\ open\ set\ in\ (Y,}\sigma {\rm ).}$
\noindent \textbf{4. CONCLUSION }In this paper we have introduced intuitionistic fuzzy $\lambda$- open\textbf{ }mappings, intuitionistic fuzzy $\lambda$-closed mappings and studied some of their properties.
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\noindent P.Rajarajeswari ([email protected]) -- Department of Mathematics, Chikkanna Government Arts College, Tirupur - 641 602, Tamil Nadu, India
\noindent G.Bagyalakshmi (g\[email protected]) -- Department of Mathematics, AJK College of Arts and Science, Coimbatore -641105. Tamil Nadu, India
\noindent
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