ON UPPER AND LOWER ALMOST CONTRA-$\mathcal{I}$-CONTINUOUS MULTIFUNCTIONS

Arivazhai, C.; Rajesh, N.; Shanthi, S. Anita

doi:10.12732/ijpam.v115i4.12

Int. J. of Pure and Appl. Math.

2017

DOI: 10.12732/ijpam.v115i4.12

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ON UPPER AND LOWER ALMOST CONTRA-$\mathcal{I}$-CONTINUOUS MULTIFUNCTIONS

C. Arivazhai¹,

N. Rajesh²,

S. Anita Shanthi³

Abstract: In this paper, we introduce and study the almost contra-I-continuous multifunctions between topological spaces and obtain some characterizations and properties of such multifunctions.

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2021

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“…The notion of upper (resp. lower) weakly I-continuous multifunction [3], when we choose α = identity operator, β = I-interior operator, δ = identity operator, θ = closure operator, and I = {∅}. That means…”

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Upper and lower $(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous multifunctions

et al. 2021

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Let $(X, \tau)$ and $(Y, \sigma)$ be topological spaces in which no separation axioms are assumed, unless explicitly stated and if $\mathcal{I}$ is an ideal on $X$.Given a multifunction $F\colon (X, \tau)\rightarrow (Y, \sigma)$, $\alpha,\beta$ operators on $(X, \tau)$, $\theta,\delta$ operators on $(Y, \sigma)$ and $\mathcal{I}$ a proper ideal on $X$. We introduce and study upper and lower $(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous multifunctions.A multifunction $F\colon (X, \tau)\rightarrow (Y, \sigma)$ is said to be: {1)} upper-$(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous if $\alpha(F^{+}(\delta(V)))\setminus \beta(F^{+}(\theta(V)))\in \mathcal{I}$ for each open subset $V$ of $Y$;\{2)} lower-$(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous if$\alpha(F^{-}(\delta(V)))\setminus \beta(F^{-}(\theta(V)))\in \mathcal{I}$ for each open subset $V$ of $Y$;\ {3)} $(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous if it is upper-\ %$(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuousand lower-$(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous. In particular, the following statements are proved in the article (Theorem 2):Let $\alpha,\beta$ be operators on $(X, \tau)$ and $\theta, \theta^{*}, \delta$ operators on $(Y, \sigma)$: \noi\ \ {1.} The multifunction $F\colon (X, \tau)\rightarrow (Y, \sigma)$ is upper $(\alpha,\beta,\theta\cap \theta^{*},\delta,\mathcal{I})$-continuous if and only if it is both upper $(\alpha,\beta,\theta,\delta,\mathcal{I})$-continuous and upper $(\alpha,\beta,\theta^{*},\delta,\mathcal{I})$-continuous. \noi\ \ {2.} The multifunction $F\colon (X, \tau)\rightarrow (Y, \sigma)$ is lower $(\alpha,\beta,\theta\cap \theta^{*},\delta,\mathcal{I})$-continuous if and only if it is both lower $(\alpha,\beta,\theta,\delta,\mathcal{I})$-continuous and lower $(\alpha,\beta,\theta^{*},\delta,\mathcal{I})$-continuous,provided that $\beta(A\cap B) =\beta(A)\cap \beta(B)$ for any subset $A,B$ of $X$.

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Upper and lower $(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous multifunctions

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ON UPPER AND LOWER ALMOST CONTRA-$\mathcal{I}$-CONTINUOUS MULTIFUNCTIONS

Abstract: In this paper, we introduce and study the almost contra-I-continuous multifunctions between topological spaces and obtain some characterizations and properties of such multifunctions.

Cited by 1 publication

References 8 publications

Upper and lower $(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous multifunctions

Upper and lower $(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous multifunctions

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