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I am reading General topology, Volume 1 By Nicolas Bourbaki. I refer to the proof of Proposition 13. Could someone kindly explain the G/H Hausdorff $\implies$ H closed part of the proof? I understand that $H$ is an equiv class for the relation $x^{-1}y \in H$ bit, but I am failing to see how the Hausdorffness relates to $H$ being closed. I am also trying to understand the converse part of the proof which I think I'd be more successful in doing so if I understand the first part first. I am trying to self-learn topology, and I apologize for the stupidness of my questions on this site. Thanks in advance.

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$G/H$ is Hausdorff implies that $G/H$ is $T_1$ implies that the singleton containing the coset $eH$ is closed and by the definition of the quotient topology, this is true if and only if its preimage under the canonical projection is closed, thus if and only if $H$ is closed in $G$.

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    Thanks, Mark. Btw what does $T_1$ mean?2011-08-19
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    @brian: It means that finite sets are closed. It's a separation property weaker than Hausdorffness (which is $T_2$).2011-08-19
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If $G/H$ is Hausdorff then every $x \in G\setminus H$ has a neighbourhood disjoint from $H$. This means $G\setminus H$ is an open set, being the union of open sets, which means that $H$ is closed.

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Let's start with the definitions. If $G/H$ is Hausdorff, then given any two distinct points, I can put open balls around them that don't intersect. Let one such point be the orbit of 1, i.e. $1\cdot H=H$, and let $gH$ be any other point. Then, I can put an open ball around $gH$ that doesn't contain $1\cdot H$. Now, you need to use the definition of quotient topology: a ball in $G/H$ is open if its preimage in $G$ is open. So I can put an open ball around $g$ that does not intersect $H$. That is one characterisation of $G\backslash H$ being open.

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    Dear Alex: What do you mean by "ball"?2011-08-19
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    @brian: If $G/H$ is Hausdorff, the point $eH$ is closed in $G/H$ and $H$ is the pre-image of that point under the canonical projection.2011-08-19
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    @Pierre-Yves: ball = open set2011-08-19
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    @Theo: Thanks! So an open ball is an open open set.2011-08-19
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    @Pierre-Yves: *touché*2011-08-19
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    @Pierre-Yves: something round, so that when you stand in the centre, you can stretch out your arms and rotate a little bit without touching the walls. That's what a neighbourhood of $gH$ looks like to me. And to you?2011-08-19
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    Dear Alex: Thanks for your answer. It seems to me that, if $a$ and $A$ are sets, you express the condition "$a\in A$" by "$A$ is a ball around $a$". I find this very confusing. Your argument is (I think) the same as Theo’s. Theo's phrasing is: "the point $eH$ is closed in $G/H$ and $H$ is the pre-image of that point under the canonical projection". Don’t you think such a sentence suffices to prove the claim?2011-08-19
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    @Pierre-Yves It does. But based on what I know about the OP's background from his previous question, I tried to phrase the whole proof as close to the basic definitions and to (my personal) intuitive viewpoints as possible.2011-08-19
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    Dear Alex: I now see that you’ve been kidding all along! Very funny! I completely fell into the trap! (Actually, I wondered if you were serious when you defined a ball as a “round thing”... But you really got me. Congatulations!)2011-08-19
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    Dear @Pierre-Yves I was actually being half serious. I find that the greatest challenge in teaching topology is to find the right balance between rigour and intuition. If one stresses geometric intuition too much, students are easily tempted to "prove" things very non-rigorously. But if one gives too little of it, the subject becomes unnecessarily dry and difficult. So in giving my answer, I was trying to strike that balance. The intuition for openness should come from metric spaces: a set is open if, no matter where one stands, one can wiggle around a little bit. Contd...2011-08-19
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    ... Closedness is less intuitive. Also, the most intuitive definition of Hausdorffness to me is that any two points can be separated away from each other by open sets. That immediately captures, why it's such a desirable property. To say that points are closed is, to me, much less intuitive. So I gave a rigorous proof, but using the most intuitive and pictorially appealing definitions, without any claim on brevity. Of course, somebody else might have completely different intuitions about all these things.2011-08-19
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    Dear Alex: [A detail: I think you agree that a topological space whose points are closed is not Hausdorff in general.] What you say makes now a lot of sense to me! I think you prove slightly more than Theo, in the sense that you don’t take for granted the fact that “Hausdorff” implies “points closed”. To do a bit of nitpicking: The work you did in $G$ to prove that $G\setminus H$ is open, had you done it in $G/H$, you would have proved *explicitly*, and with the same amount of labor, that a point of a Hausdorff space is closed (or, if you prefer, that its complement is open).2011-08-19