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Sep 21, 2017 at 6:41 vote accept Dominic van der Zypen
Sep 21, 2017 at 6:03 comment added bof @erz Seems to me any $\text T_1$ topology on $\mathbb N$ will be a refinement of your topology, e.g., a topology which is homeomorphic to the usual topology of $\mathbb Q,$ which is Hausdorff but not discrete.
Sep 21, 2017 at 5:59 answer added bof timeline score: 12
Sep 21, 2017 at 5:58 comment added erz what if $X=\mathbb{N}$ and $\tau$ be the collection of sets of the form $\{n,n+1,...\}$?
Sep 21, 2017 at 5:48 history asked Dominic van der Zypen CC BY-SA 3.0