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What are some examples of a topological space $X$ that is completely regular $T_0$ (hence Hausdorff) having a quotient space $Y = X/\sim$ that is not completely regular? Preferably such examples where $Y$ is not even regular.

Here I take "completely regular" to mean simply that each closed subset and each point not in such a subset can be separated by a numerical function; and likewise "regular" to mean that each closed subset and each point not in such a subset can be separated by neighborhoods. That is, I do not include the property of being $T_0$ as part of either definition — so that a quotient that merely fails to be $T_0$ without failing to have this separation property is not what I'm asking about,

murray
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  • Every space is a quotient of a normal $T_1$ space; see my construction here for instance. – Eric Wofsey Mar 09 '21 at 16:28
  • @EricWofsey: some examples that are more specific and concrete, where one starts with a completely regular $T_0$-space and forms a quotient of it that is not completely regular but is $T_0$ or even $T_1$? The only one I know off-hand is the quotient of the real line obtained by collapsing the set of rationals to a point. – murray Mar 11 '21 at 17:27

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