FINITE TOPOLOGICAL SPACES AND GRAPHS

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

Topological spaces associated to higher-rank graphs

We investigate which topological spaces can be constructed as topological realisations of higher-rank graphs. We describe equivalence relations on higher-rank graphs for which the quotient is again a higher-rank graph, and show that identifying isomorphic co-hereditary subgraphs in a disjoint union of two rank-k graphs gives rise to pullbacks of the associated C∗algebras. We describe a combinat...

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.