Countable Fan Tightness and Selection Games in Group-Valued Function Spaces
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913063137968128 |
|---|---|
| author | Mandal, Souvik Sarkar, Ankur |
| author_facet | Mandal, Souvik Sarkar, Ankur |
| contents | 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 $Ω$-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 is established between the $Ω$-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 Kočinac 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23671 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Countable Fan Tightness and Selection Games in Group-Valued Function Spaces Mandal, Souvik Sarkar, Ankur General Topology Primary 54C35, 54D20, 91A44, Secondary 54H11, 22A05 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 $Ω$-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 is established between the $Ω$-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 Kočinac 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. |
| title | Countable Fan Tightness and Selection Games in Group-Valued Function Spaces |
| topic | General Topology Primary 54C35, 54D20, 91A44, Secondary 54H11, 22A05 |
| url | https://arxiv.org/abs/2604.23671 |