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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University of Maragheh
2025-01-01
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| Series: | Sahand Communications in Mathematical Analysis |
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| Online Access: | https://scma.maragheh.ac.ir/article_718668_c92e329521539163f6376846656b91e6.pdf |
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| author | Devi Selvam Sivaramakrishnan Monikandan |
| author_facet | Devi Selvam Sivaramakrishnan Monikandan |
| author_sort | Devi Selvam |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-4d5a78ea4dfe4633a1df78ef942398b9 |
| institution | Kabale University |
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| language | English |
| publishDate | 2025-01-01 |
| publisher | University of Maragheh |
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| series | Sahand Communications in Mathematical Analysis |
| spelling | doaj-art-4d5a78ea4dfe4633a1df78ef942398b92025-02-11T05:28:01ZengUniversity of MaraghehSahand Communications in Mathematical Analysis2322-58072423-39002025-01-01221597410.22130/scma.2024.2006636.1380718668Recognition of Finite Bitopological Spaces with Unique Isolated PointDevi Selvam0Sivaramakrishnan Monikandan1Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli- 627 012, Tamil Nadu, India.Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli- 627 012, Tamil Nadu, India.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.https://scma.maragheh.ac.ir/article_718668_c92e329521539163f6376846656b91e6.pdfreconstructionbitopological spaceisolated pointpair-homeomorphism |
| spellingShingle | Devi Selvam Sivaramakrishnan Monikandan Recognition of Finite Bitopological Spaces with Unique Isolated Point Sahand Communications in Mathematical Analysis reconstruction bitopological space isolated point pair-homeomorphism |
| title | Recognition of Finite Bitopological Spaces with Unique Isolated Point |
| title_full | Recognition of Finite Bitopological Spaces with Unique Isolated Point |
| title_fullStr | Recognition of Finite Bitopological Spaces with Unique Isolated Point |
| title_full_unstemmed | Recognition of Finite Bitopological Spaces with Unique Isolated Point |
| title_short | Recognition of Finite Bitopological Spaces with Unique Isolated Point |
| title_sort | recognition of finite bitopological spaces with unique isolated point |
| topic | reconstruction bitopological space isolated point pair-homeomorphism |
| url | https://scma.maragheh.ac.ir/article_718668_c92e329521539163f6376846656b91e6.pdf |
| work_keys_str_mv | AT deviselvam recognitionoffinitebitopologicalspaceswithuniqueisolatedpoint AT sivaramakrishnanmonikandan recognitionoffinitebitopologicalspaceswithuniqueisolatedpoint |