Properties of Space Set Topological Spaces

Sang-Eon Han

Since a locally finite topological structure plays an important role in 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, $SST$) and further, proves that an $SST$ is an Alexandroff space satisfying the separation axiom $T_0$. Unlike a point set topology, since each element of an $SST$ is a space, the present paper names the topology by the space set topology. Besides, for a connected topological space $(X,T)$ with $|X|=2$ the axioms $T_0$, semi-$T_\frac12$ and $T_\frac12$ are proved to be equivalent to each other. Furthermore, the paper shows that an $SST$ can be used for studying both continuous and digital spaces so that it plays a crucial role in both classical and digital topology, combinatorial, discrete and computational geometry. In addition, a connected $SST$ can be a good example showing that the separation axiom semi-$T_\frac12$ does not imply $T_\frac12$.