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It is well known that a regular space is a topological space $X$ with these two properties:

1)All one point sets are closed.

2)For every $x\in X$ and every closed set $B$ (such that $x\notin B$), there exist disjoint open sets $C$ and $D$ such that:

$x\in C\quad ,\quad B\subset D$.

I am wondering how is this related to the name 'regular'? Any intuition where the name come from?

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    $\begingroup$ I don't think it's a very descriptive name. Regular means "following the rule" or standard. Clearly one of the first spaces point set topologists were considering was the real line. So it would be reasonably natural to call spaces with some properties of the real line "regular", leaving the "irregular" spaces with unusual properties. A related word in English that is similarly overused is normal. $\endgroup$
    – Anthony Quas
    Commented May 24, 2021 at 6:46
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    $\begingroup$ It's just one of those generic overused adjectives that has no real meaning except "having some desirable property". Likewise normal, perfect, proper, admissible, etc. $\endgroup$
    – Nate Eldredge
    Commented May 24, 2021 at 7:02
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    $\begingroup$ Condition 2 alone stands for "regular". Conditions 1+2 stands for $\ T_3.$ $\endgroup$
    – Wlod AA
    Commented May 24, 2021 at 7:10
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    $\begingroup$ This all reminds me of an old joke by J.-P. Serre: "regular" is a well-defined notion in mathematics -- it is so well-defined that it has about 23 different definitions. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented May 24, 2021 at 9:44
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    $\begingroup$ Some people, like the OP, define "regular" to imply $T_1$, while $T_3$ doesn't. Others, like @WlodAA use the opposite convention. Corollary 1: If you work with non-$T_1$ topologies like the Zariski topology, don't use either version of "regular". Corollary 2: Algebraic geometers can redefine "regular" to mean something totally different. $\endgroup$
    – Andreas Blass
    Commented May 24, 2021 at 15:21

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