What question did this study set out to answer?

This research aims to determine the separability properties of the function spaces related to Tychonoff spaces.

synapse

⌘+K

synapse

⌘+K

May 15, 2026Open Access

For any space X, either Cₚ (X) or CₚCₚ (X) is -separable

Key Points

This research aims to determine the separability properties of the function spaces related to Tychonoff spaces.
Establishes the existence of dense subspaces in function spaces under specific conditions.
Provides examples of spaces exhibiting non-tightness and separability characteristics.
Analyzes compact zero-dimensional spaces and their implications on functional separability.
Demonstrates that at least one of the spaces C_p(X) or C_pC_p(X) is ψ-separable for any Tychonoff space X.
Shows that for certain spaces, C_p(X) lacks a dense subspace of countable functional tightness.
Identifies conditions under which C_p(K) has a dense exponentially separable subspace if K is ω-monolithic.

Abstract

Abstract We establish that, for any Tychonoff space X, at least one of the spaces Cₚ (X) C p (X) and CₚCₚ (X) C p C p (X) has a dense subspace of countable pseudocharacter. Under MA, we give an example of a space X such that Cₚ (X) C p (X) does not have a dense subspace of countable functional tightness. We also show that there exists a compact zero-dimensional space K such that Cₚ (K, \0, 1\) C p (K, 0, 1) is exponentially separable while K is not a Sokolov space. For compact scattered spaces K of countable dispersion index, we show that Cₚ (K) C p (K) has a dense exponentially separable subspace if and only if K is ω -monolithic. Our results provide answers to several published open questions.

Mark Helpful

Bookmark

Relay

View Full Paper