Properties of Space Set Topological Spaces

Abstract

Since the locally finite topological structure can contribute to the fields of pure and applied topology,  the paper studies a special kind of locally finite spaces, so called a space set topology (for brevity, {\it SST}) and further,   proves that an  {\it SST} is an Alexandroff space satisfying the separation axiom  $T_0$.  Besides, for a topological space $(X, T)$ with $\vert X \vert =2$ the axioms $T_0$, semi-$T_{\frac 1{2}}$ and $T_{\frac 1{2}}$ are proved to be equivalent to each other.    Furthermore, the paper shows that an {\it SST} can be used for studying both continuous and digital spaces so that it plays an important role in both classical  and digital topology, combinatorial, discrete and computational geometry.  In addition, an {\it SST} can be a good example showing that the separation axiom {\it  semi-$T_{\frac 1{2}}$} does not imply  {\it $T_{\frac 1{2}}$}.