Countable networks on Malykhin's maximal topological group
Submitted: 2022-10-03
|Accepted: 2023-06-26
|Published: 2023-10-02
Copyright (c) 2023 Applied General Topology
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
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Keywords:
countable network, resolvable, linear, P-point, P-space
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Abstract:
We present a solution to the following problem: Does every countable and non-discrete topological (Abelian) group have a countable network with infinite elements? In fact, we show that no maximal topological space allows for a countable network with infinite elements. As a result, we answer the question in the negative. The article also focuses on Malykhin's maximal topological group constructed in 1975 and establishes some unusual properties of countable networks on this special group G. We show, in particular, that for every countable network N for G, the family of finite elements of N is also a network for G.
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