Low-level separation axioms for computational topology
Abstract
The present paper proves that a {\it space set topology} (for brevity, {\it SST}) is an Alexandroff space satisfying
the semi-$T_{\frac 1{2}}$-separation axiom, and it does not satisfy the pre $T_{\frac 1{2}}$-separation axiom.
This approach can play an important role in computational topology and digital geometry because an {\it SST} is not a Hausdorff space.
Next, consider an {\it SST}, denoted by $ST(X)$, over an {\it SAC} complex $C:=(X, N, dim)$.
Assume further that an element $c_j^{i_2}$ is a maximal element of $ST(X)$ (see Definition 12).
Then the present paper proves that
each singleton $\{c_j^{i_2}\}(\subset ST(X))$ is preopen and the other each singleton $\{c_j^{i}\}, c_j^{i}\neq c_j^{i_2}$ in $ST(X)$ is nowhere dense in $ST(X)$.
Finally, using an edge to edge tiling (or a face to face crystallizing) of ${\bf R}^n, n \in {\bf N}$, denoted by $T({\bf R}^n)$ as an {\it SAC} complex,
we prove that an $SST$ over $T({\bf R}^n)$ is a semi-$T_1$ space.
This approach can be also used in the fields of pure and applied mathematics as well as digital topology, computational topology and digital geometry.
In the paper we assume that given topological spaces are nonempty and connected.
the semi-$T_{\frac 1{2}}$-separation axiom, and it does not satisfy the pre $T_{\frac 1{2}}$-separation axiom.
This approach can play an important role in computational topology and digital geometry because an {\it SST} is not a Hausdorff space.
Next, consider an {\it SST}, denoted by $ST(X)$, over an {\it SAC} complex $C:=(X, N, dim)$.
Assume further that an element $c_j^{i_2}$ is a maximal element of $ST(X)$ (see Definition 12).
Then the present paper proves that
each singleton $\{c_j^{i_2}\}(\subset ST(X))$ is preopen and the other each singleton $\{c_j^{i}\}, c_j^{i}\neq c_j^{i_2}$ in $ST(X)$ is nowhere dense in $ST(X)$.
Finally, using an edge to edge tiling (or a face to face crystallizing) of ${\bf R}^n, n \in {\bf N}$, denoted by $T({\bf R}^n)$ as an {\it SAC} complex,
we prove that an $SST$ over $T({\bf R}^n)$ is a semi-$T_1$ space.
This approach can be also used in the fields of pure and applied mathematics as well as digital topology, computational topology and digital geometry.
In the paper we assume that given topological spaces are nonempty and connected.
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