arxiv: 2604.25827 · v2 · submitted 2026-04-28 · 🧮 math.GN

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Uniform homeomorphisms between C_p^*-spaces preserve pseudocompactness

Vesko Valov

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Pith reviewed 2026-05-07 13:51 UTC · model grok-4.3

classification 🧮 math.GN

keywords uniform homeomorphismsC_p^* spacespseudocompactnessTychonoff spacespointwise convergence topologybounded continuous functionskappa-pseudocompactness

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The pith

Uniform homeomorphisms between C_p^*-spaces preserve pseudocompactness of the underlying Tychonoff spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves that whenever C_p^*(X) is uniformly homeomorphic to C_p^*(Y) for Tychonoff spaces X and Y, then X is pseudocompact if and only if Y is pseudocompact. The result settles a question left open by Baars, van Mill and Tkachuk, who had already established the same preservation for linear homeomorphisms between these spaces. It extends Uspenskij's earlier theorem for the full (unbounded) function spaces C_p(X) and combines with Krupski's work to show that kappa-pseudocompactness is likewise preserved. The preservation gives a topological invariant that can be read off the function spaces without inspecting the domains directly.

Core claim

If C_p^*(X) is uniformly homeomorphic to C_p^*(Y) for Tychonoff spaces X and Y, then X is pseudocompact if and only if Y is pseudocompact. This, together with a result of Krupski, also implies that kappa-pseudocompactness is preserved under the same uniform homeomorphisms.

What carries the argument

A uniform homeomorphism between the spaces C_p^*(X) and C_p^*(Y) of bounded continuous real-valued functions equipped with the pointwise convergence topology.

If this is right

Pseudocompactness transfers in both directions under uniform homeomorphisms of the bounded function spaces.
Kappa-pseudocompactness is likewise an invariant of these uniform homeomorphisms.
The result gives a new way to compare Tychonoff spaces by examining only their bounded pointwise function spaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The preservation may extend to other compactness-type properties not yet checked under uniform rather than linear equivalences.
It raises the question whether mere (non-uniform) homeomorphisms between C_p^* spaces would still force the pseudocompactness equivalence.

Load-bearing premise

X and Y are Tychonoff spaces and the uniform homeomorphism respects the uniform structures coming from the pointwise topology on the bounded continuous functions.

What would settle it

A pair of Tychonoff spaces, exactly one of which is pseudocompact, whose C_p^* spaces are nevertheless uniformly homeomorphic would refute the claim.

read the original abstract

For any Tychonoff space $X$ let $C_p(X)$ (resp., $C^*_p(X)$) be the set of all continuous (resp., and bounded) functions on $X$ with the pointwise convergence topology. Given Tychonoff spaces $X$ and $Y$, Uspenskij \cite{us} proved that if $C_p(X)$ is uniformly homeomorphic to $C_p(Y)$, then $X$ is pseudocompact if and only if $Y$ is pseudocompact. The author and Vuma \cite{valvu} have shown that linear homeomorphisms between $C_p^*(X)$ and $C_p^*(Y)$ preserve pseudocompactness. Recently Baars-van Mill-Tkachuk \cite{bmt} gave another proof of that result and raised the question if the same remains true provided $C_p^*(X)$ and $C_p^*(Y)$ are uniformly homeomorphic. In the present paper we answer that question positively. This, together with a result of Krupski \cite{k}, implies that $\kappa$-pseudocompactness is also preserved by uniform homeomorphisms between $C_p^*$-spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Valov settles the open uniform-homeomorphism case for pseudocompactness preservation in C_p^* spaces, though only the abstract is visible here.

read the letter

Valov has given a positive answer to the open question from Baars-van Mill-Tkachuk on whether uniform homeomorphisms between C_p^* spaces preserve pseudocompactness. The note shows that if C_p^*(X) is uniformly homeomorphic to C_p^*(Y) for Tychonoff spaces X and Y, then X is pseudocompact exactly when Y is. It also combines this with Krupski's work to get the same for kappa-pseudocompactness. This builds directly on Uspenskij's earlier result for the full C_p spaces and the linear homeomorphism case that Valov and Vuma had already settled. The new piece is handling the uniform but not necessarily linear case for the bounded functions version. The paper does a clean job of stating the context and the precise extension it provides. The abstract lays out the references and the implication clearly. The main limitation right now is that we only have the abstract, with no proof steps or lemmas shown. That makes it impossible to assess the soundness of the argument or spot any potential issues in how the uniform homeomorphism is used to transfer boundedness of continuous functions. In a proof paper like this, the details matter a lot. This kind of result is mainly of interest to specialists in general topology who study function spaces with the pointwise topology and their topological invariants. A reader who has been following the preservation results under different kinds of homeomorphisms would find the resolution useful. I think it deserves a serious referee. The question was explicitly open, so confirming the answer with a full proof would be worth the time even if revisions are needed later.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove that for Tychonoff spaces X and Y, a uniform homeomorphism between C_p^*(X) and C_p^*(Y) preserves pseudocompactness, i.e., X is pseudocompact if and only if Y is. This provides a positive answer to a question of Baars-van Mill-Tkachuk, extending Uspenskij's result for C_p-spaces and prior work on linear homeomorphisms of C_p^*-spaces; combined with Krupski's theorem, it also yields preservation of κ-pseudocompactness.

Significance. If substantiated, the result would strengthen the theory of invariants for function spaces by showing that pseudocompactness (and κ-pseudocompactness) is preserved under uniform homeomorphisms of C_p^*-spaces, closing a gap between linear and uniform equivalences.

major comments (1)

The manuscript consists solely of the abstract, which states the main theorem and its consequences but supplies no proof steps, lemmas, constructions, or verification details. Without any argument showing how a uniform homeomorphism f: C_p^*(X) → C_p^*(Y) transfers the property that every continuous real-valued function on X (resp. Y) is bounded, the central claim cannot be assessed for correctness or for hidden assumptions on the Tychonoff spaces X and Y.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for their feedback on our manuscript. Below we provide a point-by-point response to the major comment.

read point-by-point responses

Referee: The manuscript consists solely of the abstract, which states the main theorem and its consequences but supplies no proof steps, lemmas, constructions, or verification details. Without any argument showing how a uniform homeomorphism f: C_p^*(X) → C_p^*(Y) transfers the property that every continuous real-valued function on X (resp. Y) is bounded, the central claim cannot be assessed for correctness or for hidden assumptions on the Tychonoff spaces X and Y.

Authors: We agree that the submitted manuscript contains only the abstract and does not include the detailed proof of the main result. This was an error in the preparation of the submission. In the revised version of the manuscript, we will incorporate the full proof, including all necessary lemmas, constructions, and verification details. This will allow the reader to see how the uniform homeomorphism preserves the pseudocompactness property for the underlying Tychonoff spaces, addressing any potential hidden assumptions. revision: yes

Circularity Check

0 steps flagged

No derivation chain present; abstract only asserts result without proof steps

full rationale

The provided document consists solely of the abstract, which states the main theorem as a positive answer to a question from Baars-van Mill-Tkachuk, extending Uspenskij's result for C_p and a prior linear-homeomorphism result by the author with Vuma, plus a consequence from Krupski. No equations, lemmas, constructions, or explicit derivation steps are supplied, so no load-bearing reductions, self-definitions, fitted predictions, or self-citation chains can be inspected or quoted. The claim is presented as extending independent prior work without any internal circular structure visible.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard definitions of Tychonoff spaces, continuous and bounded functions, uniform homeomorphisms, and pseudocompactness drawn from prior literature in general topology; no new entities or fitted parameters are introduced.

axioms (2)

standard math Tychonoff spaces are completely regular Hausdorff spaces
Invoked in the setup for C_p and C_p^* as stated in the abstract.
standard math Pseudocompactness is equivalent to every continuous real-valued function being bounded
Standard definition used throughout the abstract and referenced results.

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