Recognition of Finite Bitopological Spaces with Unique Isolated Point - Devi Selvam; Sivaramakrishnan Monikandan (Sahand Communications in Mathematical Analysis)
A mapping $ f$ from a bitopological space $(X, τ_{1}, τ_{2})$ into a bitopological space $(Y, τ^{'}_{1}, τ^{'}_{2}) $ is said to be a pair-homeomorphism if and only if the induced functions $ f_1: (X, τ_{1})→(Y, τ^{'}_{1}) $ and $ f_2: (X, τ_{2})→(Y, τ^{'}_{2}) $ are homeomorphisms. The {deck} of a bitopological space $(X, τ_{1}, τ_{2})$ 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 $(X, τ_{1}, τ_{2})$ is pair-homeomorphic to $(Y, τ^{'}_{1}, τ^{'}_{2})$. A property $\mathscr{P}$ of a bitopological space $ X $ is {recognizable} if $\mathscr{D}(X)=\mathscr{D}(Y)$ implies “$X$ has $\mathscr{P}$ if and only if $Y$ has $\mathscr{P}$”. It is shown that every finite bitopological space, with a unique isolated point and at most two non pair-homeomorphic cards, are recognizable.