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arxiv: 2601.03720 · v2 · pith:3W2ONVVInew · submitted 2026-01-07 · 🧮 math.GN · math.DS

On generalized Namioka spaces and joint continuity of functions on product of spaces

Xiongping Dai , Congying Lv , Yuxuan Xie This is my paper

Pith reviewed 2026-05-21 16:27 UTC · model grok-4.3

classification 🧮 math.GN math.DS
keywords generalized Namioka spacesjoint continuityseparately continuous functionsproduct spacestopological groupsnon-meager spacesBaire spaces
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The pith

Non-meager products of separable or pseudo-metric spaces are generalized Namioka spaces.

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

The paper defines a generalized Namioka space as one in which every separately continuous real-valued function on the product with any compact space becomes jointly continuous along at least one full slice. It shows that a product of separable spaces or of pseudo-metric spaces is such a space whenever the product itself is non-meager. The same conclusion holds for certain mixed products that are Baire or non-meager. These properties are then used to conclude that separately continuous actions of suitable right-topological groups on locally compact regular spaces, or on countably compact completely regular spaces, must be jointly continuous.

Core claim

If the product X equals the product over alpha in A of X_alpha is non-meager and each factor is separable or pseudo-metric, then X is a gN-space; if X is separable and Y pseudo-metric with X times Y Baire or non-meager then the product is an N-space or gN-space; and if G is a gN right-topological group and X locally compact regular, or if G is separable first-countable non-meager and X times X countably compact completely regular, then any separately continuous action of G on X is jointly continuous.

What carries the argument

The generalized Namioka space (gN-space): a topological space X such that for every compact Y and every separately continuous f from X times Y to the reals there exists at least one x in X making f jointly continuous at every point of the slice {x} times Y.

If this is right

  • Separately continuous actions of gN right-topological groups on locally compact regular spaces are jointly continuous.
  • Separately continuous actions of separable first-countable non-meager right-topological groups on countably compact completely regular spaces are jointly continuous.
  • Products satisfying the countable non-meager tail condition are themselves non-meager gN-spaces.
  • In products with countable pi-bases every tail set with the Baire property is either meager or residual.

Where Pith is reading between the lines

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

  • The non-meager hypothesis on products may be replaceable by weaker category conditions in some classes of spaces.
  • The joint-continuity results for actions could extend to non-locally-compact spaces if the group action satisfies additional uniformity conditions.
  • The tail-set result on Baire-property sets may combine with other selection principles to yield new automatic-continuity theorems.

Load-bearing premise

The product space must be non-meager so that a suitable point x exists at which the separately continuous function becomes jointly continuous on the slice.

What would settle it

A concrete non-meager product of separable metric spaces together with a compact Y and a separately continuous real function that fails to be jointly continuous at every slice {x} times Y.

read the original abstract

A space $X$ is called a generalized Namioka space (g$\mathcal{N}$-space), if for every compact space $Y$ and every separately continuous function $f\colon X\times Y\rightarrow\mathbb{R}$, there exists at least one point $x\in X$ such that $f$ is jointly continuous at each point of $\{x\}\times Y$. We principally prove the following results: (1) If $X=\prod_{\alpha\in A}X_\alpha$ is non-meager such that each factor is a separable space or each factor is a pseudo-metric space, then $X$ is a g$\mathcal{N}$-space. (2) If $X$ is a separable space and $Y$ a pseudo-metric space such that $X\times Y$ is Baire (resp. non-meager), then $X\times Y$ is an $\mathcal{N}$-space (resp. a g$\mathcal{N}$-space). (3) If $X=\prod_{\alpha\in A}X_\alpha$ such that each factor is separable and $\prod_{\alpha\in A^\prime}X_\alpha$ is a non-meager space for each countable subset $A^\prime$ of $A$, then $X$ is a non-meager g$\mathcal{N}$-space. (4) If $X=\prod_{\alpha\in A}X_\alpha$ such that each factor has a countable $\pi$-base, then each tail set having the property of Baire in $X$ is either meager or residual. If $G$ is a g$\mathcal{N}$ right-topological group and $X$ a locally compact regular space, or, if $G$ is a separable first countable non-meager right-topological group and $X\times X$ a countably compact completely regular space, then any separately continuous action $G\curvearrowright X$ is jointly continuous.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript defines generalized Namioka spaces (gN-spaces) and proves four main results on when product spaces X = ∏ X_α are gN-spaces (or N-spaces) under non-meager or Baire hypotheses when factors are separable, pseudo-metric, or have countable π-bases. It further derives joint continuity of separately continuous actions G ↷ X when G is a gN right-topological group and X satisfies local compactness or countable compactness conditions.

Significance. If the central claims hold, the results extend Namioka-type theorems on separate-to-joint continuity to uncountable products and supply concrete criteria for joint continuity of group actions. The explicit strengthening in result (3) via non-meager countable subproducts is a constructive feature that improves applicability.

major comments (2)
  1. [Proof of result (1)] Proof of result (1): the non-meager hypothesis on the full product X = ∏_{α∈A} X_α is invoked to locate the point x of joint continuity, yet for uncountable A the basic open sets depend on only finitely many coordinates. It is not shown explicitly how separate continuity of f, which constrains only finitely many factors at each step, combines with the global non-meager assumption to produce the required x without additional uniformity or reduction to a countable subproduct. The manuscript should supply the missing step or clarify why the argument for countable products carries over verbatim.
  2. [Statements of results (1) and (3)] Relation between results (1) and (3): result (3) adds the hypothesis that every countable subproduct is non-meager, but the text does not state whether this condition is automatically satisfied when the full product is non-meager (as assumed in (1)) or whether (1) tacitly relies on it. This relation is load-bearing for the scope of the main theorem.
minor comments (2)
  1. [Abstract] The abstract alternates between the notations “gN-space” and “g𝒩-space”; a single consistent notation should be adopted throughout the manuscript and abstract.
  2. [Result (4)] Result (4) on tail sets with the Baire property would benefit from a brief comparison with existing results on the Baire property in products with countable π-bases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight areas where the presentation of the proofs and the logical relations between our main results can be clarified. We address each major comment below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Proof of result (1)] Proof of result (1): the non-meager hypothesis on the full product X = ∏_{α∈A} X_α is invoked to locate the point x of joint continuity, yet for uncountable A the basic open sets depend on only finitely many coordinates. It is not shown explicitly how separate continuity of f, which constrains only finitely many factors at each step, combines with the global non-meager assumption to produce the required x without additional uniformity or reduction to a countable subproduct. The manuscript should supply the missing step or clarify why the argument for countable products carries over verbatim.

    Authors: We appreciate this observation on the proof of result (1). The argument relies on the fact that, in the product topology, continuity and separate continuity are determined locally by finitely many coordinates. Specifically, for a separately continuous f, the set of points where joint continuity fails can be analyzed via projections onto finite subproducts. The non-meager assumption on the full product X ensures that these 'bad' sets (which are meager in the relevant finite-coordinate subspaces) cannot cover X, allowing us to select x in the comeager set of good points. While the countable case is handled directly via Baire category theorem applications, the uncountable case carries over verbatim because no uniformity across all coordinates is required—only finite dependence at each step. We acknowledge the step was implicit rather than explicit. In the revision we will insert a clarifying paragraph (or lemma) spelling out this finite-coordinate reduction and why the non-meager hypothesis suffices without invoking countable subproducts. revision: yes

  2. Referee: [Statements of results (1) and (3)] Relation between results (1) and (3): result (3) adds the hypothesis that every countable subproduct is non-meager, but the text does not state whether this condition is automatically satisfied when the full product is non-meager (as assumed in (1)) or whether (1) tacitly relies on it. This relation is load-bearing for the scope of the main theorem.

    Authors: We clarify the logical relationship as follows. Result (1) assumes the full product X is non-meager and, under the separability or pseudo-metric hypotheses on the factors, concludes that X is a gN-space; its proof works directly with the global non-meager assumption and does not invoke or require non-meagerness of countable subproducts. Result (3), by contrast, assumes non-meagerness only of all countable subproducts (which is a weaker global hypothesis in some settings) and concludes both that X itself is non-meager and that it is a gN-space. Non-meagerness of the full product does not automatically imply non-meagerness of every countable subproduct in arbitrary spaces, but under the hypotheses of (1) the direct argument avoids needing this implication. We will add a short remark after the statements of (1) and (3) making this distinction explicit so that the scope of each theorem is clear. revision: yes

Circularity Check

0 steps flagged

No significant circularity; theorems rely on external topological assumptions

full rationale

The paper defines a gN-space and proves four results plus a corollary on joint continuity of actions. The non-meager (or Baire) hypothesis on the product or on countable subproducts is an independent premise used to locate the point x at which joint continuity holds; it is not derived from the gN conclusion or from any fitted quantity. No equations appear, no parameters are fitted to data, and no self-citation chain is invoked to justify the central claims. The derivation chain is therefore self-contained against standard Baire-category and product-topology arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard axioms of general topology (Baire category, separability, pseudo-metric properties, non-meager sets) without introducing new free parameters or invented entities.

axioms (2)
  • standard math Baire category theorem and properties of non-meager sets in product topologies
    Invoked implicitly to guarantee existence of the continuity point x in the definition of gN-spaces.
  • standard math Standard definitions of separable spaces, pseudo-metric spaces, and right-topological groups
    Used throughout the statements of results (1)-(4) and the group-action corollary.

pith-pipeline@v0.9.0 · 5900 in / 1408 out tokens · 59239 ms · 2026-05-21T16:27:08.218105+00:00 · methodology

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Reference graph

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