All Finite Topological Spaces are Weakly Reconstructible

Abstract

The deck of a topological space X is the set $\mathscr {D}(X)=\{[X-\{x\}]:x\in X\},$ where [Z] denotes the homeomorphism class of Z. A space X is topologically reconstructible if whenever $\mathscr {D}(X)=\mathscr {D}(Y)$ then X is homeomorphic to Y. A topological space X is said to be weakly reconstructible if it is reconstructible from its multi-deck. It is shown that all finite topological spaces are weakly reconstructible.