Show that topologies are not determined by their convergent sequences

Asked Oct 16 '22 at 15:19

Active Oct 16 '22 at 15:37

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German: Topologien werden also durch ihre Filterkonvergenz determiniert. Mit Folgen klappt das einfach nicht mehr. Sei $X$ überzählbar. Sowohl in der diskreten Topologie als auch in der kofiniten Topologie auf $X$ konvergieren lediglich die Folgen, die eventuell konstant sind, und sie konvergieren nur gegen die jeweilige Konstante. Trotzdem sind beide Topologien verschieden. 1:0 für die Filter! - Source

Translation by Me: Topologies are hence determined by their filter convergence. With sequences that doesn't happen any more. Let $X$ be an uncountable set. In both the discrete and cofinite topology only the sequences converge which are eventually constant and they converge only to their respective constants. Despite of that, both the topologies are different. 1:0 for the filter.

Has the Author made a mistake here? In this post on MSE, it is said that if sequence is only of different terms then everything is a limit but the other says that in Discrete and Cofinite we have the same convergent sequences.

edited Oct 16 '22 at 15:37

Paul Frost

87,968

asked Oct 16 '22 at 15:19

Clemens Bartholdy

14,222

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It must be a typo. My guess is that the author meant to write cocountable topology, which has the same convergent sequences as the discrete topology. – José Carlos Santos Oct 16 '22 at 15:25
I don't catch at all the meaning, in your last sentence of "1:0 for the Filter." – Jean Marie Oct 16 '22 at 15:30
Hmm so it's like one positive point for the filter. You know like football they give one scorenpoint for team which makes goal @Jean Marie – Clemens Bartholdy Oct 16 '22 at 15:35
Oh, I see. I should have guessed ! – Jean Marie Oct 16 '22 at 15:44
Thanx @JoséCarlosSantos – Clemens Bartholdy Oct 16 '22 at 15:54

Show that topologies are not determined by their convergent sequences

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