A Topology with Order, Graph and an Enumeration Problem

M. Rostami

Abstract: A possible topologies on a two-point set $S=\{x,y\}$ are, indiscreet, discrete and two copies of $\{\phi,\{x\},S\}$, a topology on $S$, called the {\sl Sierpinski topology}.
Craveiro de Carvalho and d'Azevedo Breda [1] call a topological space $X$ {\sl locally Sierpinski space}, if every point $x\in X$ has a neighbourhood homeomorphic to the Sierpinski space. (Remember the definition of a topological manifold). The authors then have established there some properties of locally Sierpinski spaces, but left the following enumeration problem open: Let $X$ be a finite set with $n$ elements. Find (up to homeomorphism) the number of different topologies making $X$ into a Locally Sierpinski space. In this note we use a well-known ordering of a topological space and its corresponding diagram to solve this enumeration problem.

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