Recognition of Finite Bitopological Spaces with Unique Isolated Point
A mapping $ f$ from a bitopological space $(X, \tau_{1}, \tau_{2})$ into a bitopological space $(Y, \tau^{'}_{1}, \tau^{'}_{2}) $ is said to be a pair-homeomorphism if and only if the induced functions $ f_1: (X, \tau_{1})\rightarrow(Y, \tau^{'}_{1}) $ and $ f_2: (X, \tau_{2})\rightar...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Maragheh
2025-01-01
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| Series: | Sahand Communications in Mathematical Analysis |
| Subjects: | |
| Online Access: | https://scma.maragheh.ac.ir/article_718668_c92e329521539163f6376846656b91e6.pdf |
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| Summary: | A mapping $ f$ from a bitopological space $(X, \tau_{1}, \tau_{2})$ into a bitopological space $(Y, \tau^{'}_{1}, \tau^{'}_{2}) $ is said to be a pair-homeomorphism if and only if the induced functions $ f_1: (X, \tau_{1})\rightarrow(Y, \tau^{'}_{1}) $ and $ f_2: (X, \tau_{2})\rightarrow(Y, \tau^{'}_{2}) $ are homeomorphisms. The {deck} of a bitopological space $(X, \tau_{1}, \tau_{2})$ is the set $\mathscr{D}(X)=\{[X_{x}]:x\in X\},$ where $[Z]$ denotes the pair-homeomorphism class of $Z$. A bitopological space $X$ is {reconstructible} if whenever $\mathscr{D}(X)=\mathscr{D}(Y)$ then $(X, \tau_{1}, \tau_{2})$ is pair-homeomorphic to $(Y, \tau^{'}_{1}, \tau^{'}_{2})$. A property $\mathscr{P}$ of a bitopological space $ X $ is {recognizable} if $\mathscr{D}(X)=\mathscr{D}(Y)$ implies \textquotedblleft$X$ has $\mathscr{P}$ if and only if $Y$ has $\mathscr{P}$\textquotedblright. It is shown that every finite bitopological space, with a unique isolated point and at most two non pair-homeomorphic cards, are recognizable. |
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| ISSN: | 2322-5807 2423-3900 |