Talk:Fréchet–Urysohn space

Incorrect paragraph in the lead
Fréchet–Urysohn spaces are the most general class of spaces for which sequences suffice to determine all topological properties of subsets of the space. That is, Fréchet–Urysohn spaces are exactly those spaces for which knowledge of which sequences converge to which limits (and which sequences do not) suffices to completely determine the space's topology.: This is incorrect.

The most general class of spaces in question is the class of sequential spaces and not the Frechet-Urysohn spaces, since knowing which sequences converge to which limits determines the sequentially closed (or open) sets, hence the closed (or open) sets in a sequential space, that is, it uniquely determines the topology in the sequential case. PatrickR2 (talk) 05:08, 3 June 2023 (UTC)


 * The misleading claims have now been removed. PatrickR2 (talk) 22:37, 17 January 2024 (UTC)

Sequential and Fréchet–Urysohn
A more direct way to see that Fréchet–Urysohn implies sequential would be using the following characterizations

Sequential space: $$\operatorname{scl}_X^{(\omega_1)} S = \operatorname{cl}_X S$$ for every subset $$S \subseteq X;$$

Fréchet–Urysohn space: $$\operatorname{scl}_X S = \operatorname{cl}_X S$$ for every subset $$S \subseteq X.$$ 129.104.241.214 (talk) 16:56, 10 March 2024 (UTC)