Intereting Posts

Spectrum of finite $k$-algebras
Proof that a certain derivation is well defined
How many turns can a chess game take at maximum?
Simple algorithm for generating Poisson distribution
Compute $1^2 + 3^2+ 5^2 + \cdots + (2n-1)^2$ by mathematical induction
A question with infinity
Find the Fourier transform of $\frac1{1+t^2}$
How to show the series $\displaystyle\sum_{\xi\in\mathbb Z^n}\frac{1}{(1+|\xi|^2)^{s/2}}$ converges if and only if $s>n$?
Roll an N sided die K times. Let S be the side that appeared most often. What is the expected number of times S appeared?
Cutting the Cake Problem
Counting strings containing specified appearances of words
Calculating a Lebesgue integral involving the Cantor Function
A positive polynomial is the sum of two squares in $\mathbb{R}$
Are there real-life relations which are symmetric and reflexive but not transitive?
I have to show $(1+\frac1n)^n$ is monotonically increasing sequence

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.

- When a covering map is finite and connected, there exists a loop none of whose lifts is a loop.
- If M is a non-orientable closed connected 3 manifold prove H1(M) is an infinite group.
- In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$
- Given a fiber bundle $F\to E\overset{\pi}{\to} B$ such that $F,B$ are compact, is $E$ necessarily compact?
- How can I prove that two sets are isomorphic, short of finding the isomorphism?
- Does an uncountable discrete subspace of the reals exist?
- Dense subset of $C(X)$
- $Y$ is $T_1$ iff there is regular space $X$ s.t. all continuous function from $X$ to $Y$ is constant
- Continuous image of a locally connected space which is not locally connected
- Which “limit of ultrafilter” functions induce a compact Hausdorff topological structure?

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.

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.

$G/H$ 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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- Fermats Little Theorem
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- Induction Inequality Proof with Product Operator $\prod_{k=1}^{n} \frac{(2k-1)}{2k} \leq \frac{1}{\sqrt{3k+1}}$
- How to find non-cyclic subgroups of a group?
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