Timeline for Countable network vs countable Borel network

8 events

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Jun 15, 2020 at 7:27	history	edited	CommunityBot		Commonmark migration
Jan 16, 2019 at 14:35	history	edited	Taras Banakh	CC BY-SA 4.0	Added a Remark in Edit
Jan 9, 2019 at 16:52	comment	added	Taras Banakh		@JamesHanson A Hausdorff example can be constructed as follows: Let $\tau_E$ be the standard Euclidean topology on the real line and $\tau$ be the topology generated by the subbase $\tau_E\cup\{\mathbb R\setminus \mathbb Q\}$. The real line endowed with the topology $\tau$ is a functionally Hausdorff second-countable space, which does not have a countable closed network (to prove this fact, apply the Baire Theorem).
Jan 9, 2019 at 16:46	history	edited	Taras Banakh	CC BY-SA 4.0	deleted 1 character in body
Jan 9, 2019 at 16:38	comment	added	Taras Banakh		@JamesHanson The real line endowed with the cofinite topology has a countable network but does not have a countable closed network (since the closure of any infinite set is the whole line). This is a $T_1$-example, but it is not Hausdorff.
Jan 9, 2019 at 16:33	comment	added	Taras Banakh		@JamesHanson The two-point space $X=\{a,b\}$ with topology $\{\emptyset,\{a\},X\}$ has a countable open network but does not have countable closed network. But it is not $T_1$-space. Now I will think on a $T_1$-example.
Jan 9, 2019 at 16:12	comment	added	James E Hanson		Do you have an example of a space with a countable network but not a countable closed network?
Jan 9, 2019 at 14:40	history	asked	Taras Banakh	CC BY-SA 4.0