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arxiv: 1907.05153 · v1 · pith:UHOTOLWJnew · submitted 2019-07-11 · 🧮 math.GN

On Ramsey properties, function spaces, and topological games

Steven Clontz , Alexander V. Osipov This is my paper

Pith reviewed 2026-05-24 22:40 UTC · model grok-4.3

classification 🧮 math.GN
keywords selective separabilitytopological gamesRamsey propertiesfunction spacescovering propertiesC_p(X)sequential properties
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The pith

Strategically selectively separable T3 spaces without isolated points are Markov selectively separable.

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

The paper takes up Gruenhage's open question on whether strategically selectively separable spaces are always Markov selectively separable. It records that a strong form of the implication, in which the second player picks single points rather than finite sets, follows at once for every T3 space without isolated points. Parallel results are obtained for selective sequential separability under an added Ramsey assumption, and covering properties on a space X are shown to match sequential properties on the function space C_p(X).

Core claim

As a corollary of a theorem of Berner and Juhász, every T3 space without isolated points that is strategically selectively separable is Markov selectively separable when the second player is restricted to single-point moves. When the Ramsey property is present, strong selective sequential separability reduces to a weaker condition on any countable sequentially dense subset. The γ- and ω-covering properties on X are equivalent to the corresponding sequential covering properties on C_p(X). A strengthening of the Ramsey property is defined that remains equivalent to the α2 and α4 properties in the setting of C_p(X).

What carries the argument

The selective separability games together with the Ramsey property, which reduce game outcomes on the whole space to conditions on a countable dense subset.

If this is right

  • The γ- and ω-covering properties on X become equivalent to the corresponding sequential properties on C_p(X).
  • Under the Ramsey property, strong selective sequential separability reduces to a condition on a countable sequentially dense subset.
  • A strengthened Ramsey property remains equivalent to α2 and α4 when working inside C_p(X).
  • Results known for countable spaces extend to the larger class of T3 spaces without isolated points.

Where Pith is reading between the lines

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

  • The same game equivalences may hold in some non-T3 spaces, but the paper supplies no evidence either way.
  • The reduction via Ramsey property suggests that countable dense subsets control many selection games once the Ramsey assumption is granted.
  • One could check whether the covering-property equivalences survive when C_p(X) is replaced by other function-space topologies.

Load-bearing premise

The space is T3 and has no isolated points.

What would settle it

A concrete T3 space with no isolated points that is strategically selectively separable yet fails to be Markov selectively separable under single-point moves.

read the original abstract

An open question of Gruenhage asks if all strategically selectively separable spaces are Markov selectively separable, a game-theoretic statement known to hold for countable spaces. As a corollary of a result by Berner and Juh$\acute{a}$sz, we note that the strong version of this statement, where the second player is restricted to selecting single points in the rather than finite subsets, holds for all $T_3$ spaces without isolated points. Continuing this investigation, we also consider games related to selective sequential separability, and demonstrate results analogous to those for selective separability. In particular, strong selective sequential separability in the presence of the Ramsey property may be reduced to a weaker condition on a countable sequentially dense subset. Additionally, $\gamma$- and $\omega$- covering properties on $X$ are shown to be equivalent to corresponding sequential properties on $C_p(X)$. A strengthening of the Ramsey property is also introduced, which is still equivalent to $\alpha_2$ and $\alpha_4$ in the context of $C_p(X)$.

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

0 major / 3 minor

Summary. The manuscript addresses Gruenhage's question on whether every strategically selectively separable space is Markov selectively separable. As a corollary of Berner-Juhász, it establishes that the strong form (where the second player selects single points) holds for all T3 spaces without isolated points. It develops parallel results for selective sequential separability, showing that the strong version reduces to a condition on a countable sequentially dense subset when the Ramsey property holds. It proves equivalences between the γ- and ω-covering properties of X and corresponding sequential properties of C_p(X), and introduces a strengthening of the Ramsey property that remains equivalent to α2 and α4 for C_p(X).

Significance. If the stated corollaries and equivalences hold, the work advances the theory of selection principles and topological games by linking game-theoretic separability notions to covering properties on function spaces. The explicit reduction under the Ramsey property and the new strengthening of that property (still equivalent to the classical α-properties) constitute concrete technical contributions that can be checked against the cited prior results of Berner-Juhász.

minor comments (3)
  1. [§1] §1, line after the statement of the main corollary: the parenthetical remark on 'strong version' should explicitly recall the definition of the strong game (single-point moves) to avoid forcing the reader to consult the earlier literature.
  2. [Theorem on covering equivalences] The equivalence statements for γ- and ω-covering properties (presumably Theorem 3.4 or 4.2) are stated only for T3 spaces; it is unclear whether the proofs use regularity or merely Hausdorffness, and a short remark on the minimal separation axiom would clarify the scope.
  3. [Abstract] The abstract contains the LaTeX fragment 'Juh$acute{a}$sz'; this should be rendered uniformly as 'Juhász' throughout the text and references.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work on Gruenhage's question and related selection principles. The recommendation of minor revision is noted. No specific major comments were provided in the report, so we have no points requiring point-by-point rebuttal at this stage and will incorporate any minor editorial changes as needed.

Circularity Check

0 steps flagged

No significant circularity; results rest on external corollary and independent equivalences

full rationale

The central claims are explicitly framed as corollaries to the external result of Berner and Juhász (distinct authors) for the selective separability statement under T3 + no isolated points, with new arguments supplied for the sequential versions and the equivalences between γ-/ω-covering properties on X and corresponding sequential properties on C_p(X). The reduction for strong selective sequential separability is conditioned on the Ramsey property as an explicit assumption rather than derived internally. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain; the paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard axioms of general topology (T3 separation, no isolated points) and prior theorems; no free parameters, new entities, or ad hoc assumptions are introduced in the abstract.

axioms (2)
  • standard math T3 (regular) separation axiom and absence of isolated points
    Invoked for the corollary on selective separability from Berner and Juhász.
  • domain assumption Ramsey property for spaces
    Used as the condition under which strong selective sequential separability reduces to a weaker condition on a countable subset.

pith-pipeline@v0.9.0 · 5710 in / 1246 out tokens · 26889 ms · 2026-05-24T22:40:56.281745+00:00 · methodology

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Reference graph

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