Type: Article

On a construction of Malykhin

Martínez-Ranero, C.A., Ramos-Garcia, U. A.;

Abstract:

In [Uspekhi Mat. Nauk 34 (1979), no. 6(210), 59–66; MR0562819], V. I. Malykhin constructed by forcing a nondiscrete Hausdorff extremally disconnected topological group which is linear (the neutral element has a neighborhood base consisting of subgroups) in which all countable subsets are closed and discrete. In this paper the authors present a construction with similar ideas, but using the diamond principle instead of forcing and obtaining a group of cardinality ?1. The main result is the following: Theorem. ? implies that there exists a nondiscrete Hausdorff extremally disconnected linear group topology on G=([?1]<?0,?) in which all countable subsets are closed and discrete. Here, ? denotes the symmetric difference, and ? is Jensen's diamond principle. As a corollary of the construction, a countable nondiscrete Hausdorff extremally disconnected group is also obtained (as a quotient group of G).