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Mar 24, 2020 at 12:58 comment added მამუკა ჯიბლაძე @Ante This will not change much: for each subset $Y$ there is a topology consisting of those sets which contain $Y$ (together with the empty set). These topologies are all different for different $Y$, and for infinite $Y$ they consist of infinite sets only (except for the empty set, which should be excluded by you anyway).
Mar 24, 2020 at 7:57 comment added user153451 Is much changed if every topology consists of an infinite number of sets such that each of them is also infinite?
Mar 24, 2020 at 4:37 history answered Keith Kearnes CC BY-SA 4.0