## Abstract

In 1976, Nickolas showed that for each natural n, the free topological group F(X^{n}) is topologically isomorphic to a subgroup of F(X) provided X is a compact space or, more generally, a k_{ω}-space. We complement the Nickolas’ embedding theorem by showing that it remains true for every topological space X such that all finite powers of X are pseudocompact. For example, all pseudocompact k-spaces enjoy this property. Also, we extend the embedding theorem to the class of NC_{ω}-spaces that includes, in particular, the k_{ω}-spaces and the well-ordered spaces of ordinals [0,α), for every ordinal α. Our results are quite sharp because we present a first example of a Tychonoff space Z such that F(Z) does not contain an isomorphic copy of the group F(Z^{2}). In addition, our space Z is countably compact, separable, and its square Z^{2} is not pseudocompact.

Original language | American English |
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Article number | 87 |

Journal | Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas |

Volume | 118 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jul 2024 |

## Keywords

- 22A05 (primary)
- 54C45
- 54D20 (secondary)
- C-embedding
- Countably compact space
- Free topological group
- Pseudocompact space

## All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Geometry and Topology
- Computational Mathematics
- Applied Mathematics