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Functional countability and exponential separability of product spaces and subspaces

Rodrigo Hernández-Gutiérrez, Santi Spadaro

Abstract

We investigate the behavior of functional countability and exponential separability in products and subspaces of topological spaces. We solve a problem of Tkachuk by showing that the product of functionally countable pseudocompact spaces is itself functionally countable. Solving another problem of Tkachuk, we show that it is independent of ZFC whether regular spaces which have all their subspaces functionally countable are hereditarily Lindelöf. Finally, we prove that the $σ$-product of non-zero ordinals is exponentially separable, thereby extending a result of Kemoto and Szeptycki.

Functional countability and exponential separability of product spaces and subspaces

Abstract

We investigate the behavior of functional countability and exponential separability in products and subspaces of topological spaces. We solve a problem of Tkachuk by showing that the product of functionally countable pseudocompact spaces is itself functionally countable. Solving another problem of Tkachuk, we show that it is independent of ZFC whether regular spaces which have all their subspaces functionally countable are hereditarily Lindelöf. Finally, we prove that the -product of non-zero ordinals is exponentially separable, thereby extending a result of Kemoto and Szeptycki.
Paper Structure (4 sections, 15 theorems, 6 equations)

This paper contains 4 sections, 15 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Products of pseudocompact spaces
  3. Finite products and sigma-products of ordinals
  4. Hereditarily Lindelöf spaces

Key Result

Theorem 2

tka-nice_subclass A space $X$ is functionally countable if, for every countable family $\mathcal{F}$ of zero-sets, there is a countable subset $A$ of $X$ that is strongly dense in $\mathcal{F}$.

Theorems & Definitions (29)

  • Definition 1
  • Theorem 2
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • proof
  • Proposition 8
  • proof
  • Theorem 9
  • ...and 19 more