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The Kirch space is topologically rigid. (English) Zbl 1479.54043

The Golomb space (resp. the Kirch space) is the set \(\mathbb{N}\) endowed with the topology generated by the base consisting of \(a+b(\mathbb{N}\cup \{0\})\) where \(a,b\in \mathbb{N}\) and \(b\) is a (square-free) number, coprime with \(a\). The Golomb space is connected but not locally connected, whereas the Kirch space is connected and locally connected. Recently T. Banakh et al. [Commentat. Math. Univ. Carol. 62, No. 3, 347–360 (2021; Zbl 07442496)] proved that the Golomb space is topologically rigid, i.e. it has trivial homeomorphism group. In the article under review it is shown that also the Kirch space is topologically rigid. As an important step in the proof it is shown that the Kirch space is superconnected, i.e. for any \(n\in\mathbb{N}\) and nonempty open sets \(U_1,\dots ,U_n\) the intersection \(\bar{U_1},\dots ,\bar{U_n}\) is not empty.
Reviewer: Hans Weber (Udine)

MSC:

54D05 Connected and locally connected spaces (general aspects)
11A41 Primes

Citations:

Zbl 07442496

Software:

OEIS
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Full Text: DOI arXiv

References:

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