Topological spaces with a $\sigma $-point finite base

Finite Topological Spaces

(1) Let A be a set and f be a finite sequence of elements of 2A. Suppose that for every natural number i such that 1 i and i < len f holds πi f πi+1 f : Let i, j be natural numbers. If i j and 1 i and j len f ; then πi f π j f : (2) Let A be a set and f be a finite sequence of elements of 2A. Suppose that for every natural number i such that 1 i and i< len f holds πi f πi+1 f : Let i, j be natu...

Finite Topological Spaces

The Euler Characteristic of Finite Topological Spaces

The purpose of this paper is to illustrate the relationship between the topological property of the Euler characteristic and a combinatorial object, the Möbius function, in the context of finite T0-spaces. To do this I first explain the fundamental connection between such spaces and finite partially ordered sets by proving some facts fundamental to the study of finite spaces. Then I define the ...

Persistent Homology of Finite Topological Spaces

We introduce homology and finite topological spaces. From the basis of that introduction, persistent homology is applied to finite spaces. We prove an equivalence between persistent homology and normal homology in the context of finite topological spaces and introduce an extended pseudometric on finite topological spaces, using the results of Minian.

P-persistent homology of finite topological spaces

Let P be a finite partially ordered set, briefly a finite poset. We will show that for any P -persistent object X in the category of finite topological spaces, there is a P− weighted graph, whose clique complex has the same P -persistent homology as X.