Upper and lower $(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous multifunctions
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...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | German |
| Published: |
Ivan Franko National University of Lviv
2021-06-01
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| Series: | Математичні Студії |
| Subjects: | |
| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/25 |
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| Summary: | 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-\ %$(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous
and 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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| ISSN: | 1027-4634 2411-0620 |