Equalities for the extent of infinite products and Σ-products

Yasushi Hirata, Toshimichi Usuba, Yukinobu Yajima*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


For a space X, let e(X)=ω⋅sup{|D|:D is a closed discrete subset in X}, which is called the extent of X. Here we deal with the following two questions: (1) For a product space X=∏λ∈ΛXλ, when is e(X)=|Λ|⋅sup{e(Xλ):λ∈Λ}? (2) For a Σ-product Σ of spaces Xλ,λ∈Λ, when is e(Σ)=sup{e(Xλ):λ∈Λ}? We show that the equalities in these questions hold if each Xλ is a strict p-space or a strong Σ-space and, in the case of the first question, if the cardinality of the index set Λ is less than the first weakly inaccessible. For semi-stratifiable spaces, we show that a slightly weaker form of these equalities holds.

Original languageEnglish
Article number107946
JournalTopology and its Applications
Publication statusPublished - 2022 Feb 15


  • Extent
  • p-Space
  • Product
  • Semi-stratifiable space
  • Strict p-space
  • Strong β-space
  • Strong Σ-space
  • Σ-product

ASJC Scopus subject areas

  • Geometry and Topology


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