arxiv: 2604.23671 · v1 · submitted 2026-04-26 · 🧮 math.GN

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Countable Fan Tightness and Selection Games in Group-Valued Function Spaces

Ankur Sarkar, Souvik Mandal

Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:49 UTC · model grok-4.3

classification 🧮 math.GN

keywords selection principlestopological gamesfunction spacesfan tightnessMenger gameRothberger gameC_p(X)topological groups

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The pith

Player II wins the Ω-Menger game on a Tychonoff space X exactly when it wins the countable fan tightness game on C_p(X, G) at the identity, for suitable groups G.

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

The paper proves two equivalences between games: Player II has a winning strategy in the Ω-Menger game on X if and only if it has one in the countable fan tightness game on the function space C_p(X, G) evaluated at the identity map. An analogous equivalence links the Ω-Rothberger game on X to the countable strong fan tightness game on the same function space. These hold when G is any non-trivial metrizable arc-connected topological group. The results generalize earlier characterizations that were restricted to real-valued functions and move selection-principle equivalences into the setting of explicit strategies.

Core claim

Player II has a winning strategy in the Ω-Menger game on X if and only if Player II has a winning strategy in the countable fan tightness game on C_p(X, G) at the identity function. The analogous equivalence holds between the Ω-Rothberger game on X and the countable strong fan tightness game on C_p(X, G) at the identity function.

What carries the argument

The countable fan tightness game (and its strong variant) played on the group-valued function space C_p(X, G) at the identity element, shown to be equivalent via winning strategies to the Ω-Menger and Ω-Rothberger games on the domain X.

If this is right

The game-theoretic tightness properties of C_p(X, G) are independent of the particular choice of G.
These tightness properties are preserved when two function spaces are G-equivalent.
The equivalences continue to hold when players are restricted to Markov strategies.

Where Pith is reading between the lines

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

The same strategy-transfer technique might apply to other tightness games or selection principles not covered here.
It would be natural to check whether the equivalences survive when G is allowed to be non-metrizable or non-arc-connected.
The independence from G suggests one could study these properties by fixing a convenient group such as the circle or a Lie group.

Load-bearing premise

G must be a non-trivial metrizable arc-connected topological group.

What would settle it

A Tychonoff space X together with a qualifying group G such that Player II possesses a winning strategy in the Ω-Menger game on X yet lacks one in the countable fan tightness game on C_p(X, G) at the identity (or vice versa).

read the original abstract

Game-theoretic characterizations of selection principles provide a powerful framework for analyzing covering properties through strategic interactions. For a Tychonoff space $X$ and a non-trivial metrizable arc-connected topological group $G$, we prove that Player~II has a winning strategy in the $\Omega$-Menger game on $X$ if and only if Player~II has a winning strategy in the countable fan tightness game on $C_p(X, G)$ at the identity function. The analogous equivalence is established between the $\Omega$-Rothberger game on $X$ and the countable strong fan tightness game on $C_p(X, G)$ at the identity function. These results extend the game-theoretic characterizations of Clontz from $G = \mathbb{R}$ to arbitrary metrizable arc-connected groups, and lift the selection-principle equivalences of Ko\v{c}inac to the game-theoretic setting. As consequences, we establish that the game-theoretic tightness properties of $C_p(X,G)$ are independent of $G$, preserved under $G$-equivalence, and remain valid for Markov strategies.

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

This extends Clontz's game equivalences from real functions to general metrizable arc-connected groups and records some independence and preservation consequences.

read the letter

The main thing here is that Mandal and Sarkar take the game characterizations Clontz gave for C_p(X) with real values and carry them over to C_p(X, G) when G is any non-trivial metrizable arc-connected topological group. They show Player II has a winning strategy in the Ω-Menger game on X if and only if the same holds for the countable fan tightness game on the function space at the identity, and likewise for the Ω-Rothberger game and countable strong fan tightness. They also move some of Kočinac's selection-principle equivalences into the game setting and note that the resulting tightness properties do not depend on which G you pick, survive G-equivalence, and still work when strategies are restricted to Markov ones. The proofs go through the pointwise topology and the group operations, using arc-connectedness mainly to lift paths for the strategy responses. Once the equivalences are in place the independence and preservation statements drop out directly, so it is useful to have them written down rather than left as remarks. The conditions on G are stated clearly up front, so the reach of the results is not overstated. The argument looks direct with no obvious circularity or hidden fitting, and the stress-test finds no load-bearing flaw in the reduction steps. A referee would still want to check the concrete strategy constructions for any edge cases with the group structure. This is aimed at people already working on selection principles and games in function spaces. A reader who knows the Clontz and Kočinac papers will see the value in the explicit generalizations and the listed corollaries. It shows straightforward thinking and honest engagement with the prior literature. I would bring it to a reading group to talk through how the group operations interact with the games. I would not cite it in my own work in the next year unless I were extending this exact line, but the paper is solid enough to deserve a serious referee.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that for a Tychonoff space X and non-trivial metrizable arc-connected topological group G, Player II has a winning strategy in the Ω-Menger game on X if and only if Player II has a winning strategy in the countable fan tightness game on C_p(X, G) at the identity function. An analogous equivalence holds between the Ω-Rothberger game on X and the countable strong fan tightness game on C_p(X, G) at the identity. These extend Clontz's game characterizations from G = ℝ to general such groups and lift Kočinac's selection-principle equivalences to the strategic setting. Consequences include independence of the game-theoretic tightness properties from G, preservation under G-equivalence, and validity for Markov strategies.

Significance. If the derivations hold, the work provides a robust generalization showing that the game equivalences are independent of the specific choice of G within the stated class. This strengthens the connection between selection principles on X and tightness properties in C_p(X, G), extends prior results to arbitrary metrizable arc-connected groups, and confirms validity under Markov strategies, offering a more unified framework for analyzing covering properties via games in function spaces.

minor comments (2)

[Abstract] The abstract refers to 'G-equivalence' without a brief definition or forward reference; this should be clarified in the introduction for readers unfamiliar with the term.
[Main results] In the strategy constructions, the use of arc-connectedness for path-lifting is invoked but could benefit from an explicit remark on why metrizability alone is insufficient for the general case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We are grateful to the referee for their detailed summary of our work and for recommending a minor revision. The referee correctly identifies the key results, including the extensions of Clontz's characterizations to general metrizable arc-connected groups and the lifting to strategic settings. Since the report does not include any specific major comments or criticisms, we do not have individual points to address. We believe the manuscript is in good shape and are happy to incorporate any minor suggestions from the editor.

Circularity Check

0 steps flagged

No significant circularity; direct extensions of independent prior results

full rationale

The paper derives game equivalences between selection principles on X and fan-tightness properties on C_p(X,G) via explicit strategy constructions that reduce one side to the other using the pointwise topology and the group structure of G. These steps cite and extend results from Clontz (for G=R) and Kočinac (selection principles) without self-citation chains, without redefining inputs as predictions, and without importing uniqueness theorems from the authors' prior work. The arc-connectedness hypothesis on G is stated upfront and invoked only for path-lifting in the strategy responses, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claims rest on standard domain assumptions of topology and games; no free parameters or invented entities appear.

axioms (2)

domain assumption X is a Tychonoff space
Standard for defining C_p(X, G)
domain assumption G is non-trivial metrizable arc-connected topological group
Required for the stated equivalences

pith-pipeline@v0.9.0 · 8945 in / 1218 out tokens · 62843 ms · 2026-05-08T04:49:09.457843+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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