The Kirch space is topologically rigid

TL;DR

This paper proves that the Kirch space, a topological space based on positive integers with a specific topology, is topologically rigid, meaning it has no nontrivial homeomorphisms, extending known results from the Golomb space.

Contribution

The paper establishes the topological rigidity of the Kirch space, a previously unproven property for this space, building on recent results about similar spaces.

Findings

Kirch space is topologically rigid.

Kirch space has trivial homeomorphism group.

Extends rigidity results from Golomb space to Kirch space.

Abstract

The $Golomb$ $space$ (resp. the $Kirch$ $space$) is the set $\mathbb N$ of positive integers endowed with the topology generated by the base consisting of arithmetic progressions $a+b\mathbb N_0=\{a+bn:n\ge 0\}$ where $a\in\mathbb N$ and $b$ is a (square-free) number, coprime with $a$. It is known that the Golomb space (resp. the Kirch space) is connected (and locally connected). By a recent result of Banakh, Spirito and Turek, the Golomb space has trivial homeomorphism group and hence is topologically rigid. In this paper we prove the topological rigidity of the Kirch space.