Embedding the free topological group F(Xn) into F(X)

Arkady G. Leiderman, Mikhail G. Tkachenko

Research output: Contribution to journalArticlepeer-review


In 1976, Nickolas showed that for each natural n, the free topological group F(Xn) is topologically isomorphic to a subgroup of F(X) provided X is a compact space or, more generally, a kω-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 NCω-spaces that includes, in particular, the kω-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(Z2). In addition, our space Z is countably compact, separable, and its square Z2 is not pseudocompact.

Original languageEnglish
Article number87
JournalRevista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
Issue number3
StatePublished - 1 Jul 2024


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

ASJC Scopus subject areas

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


Dive into the research topics of 'Embedding the free topological group F(Xn) into F(X)'. Together they form a unique fingerprint.

Cite this