Closed subgroup on a topological group

Question

Today a student ask me the following question regarding topological groups in the tutorial centre.

Let $H$ be a subgroup of a topological group $G$. Suppose that $U$ is an open containing the identity such that $\overline{U} \cap H$ is closed. Prove that $H$ is closed.

I am a bit rusty on point set topology so I couldn't figure out a proper solution. I tried a straightforward approach without success:

Let $x \in \overline{H}$, we want to show that $x \in H$. Since $e \in U$ then $x \in xU$, and we can take an element $y \in xU \cap H$, therefore $y = xa = h$ for some $a \in U$, $h\in H$, so $x = ha^{-1}$

Trying to prove that $a^{-1} \in H$, I want to use somehow that $\overline{U} \cap H$ is closed; but I'm stuck at this point. I wish I could dedicate more time to solve the problem and review the basic properties of topological groups, but unfortunely I am overhelmed with my own stuff.

Answer 1

Suppose $x\in\overline{H}$. Let $V$ be a neighborhood of $1$ such that $VV^{-1}\subseteq U$. Since $Vx$ is an open neighborhood of $X$, $x\in\overline{H\cap Vx}$. Now pick some $y\in H\cap Vx$ and multiply everything by $y^{-1}$ on the right. We conclude that $xy^{-1}\in\overline{Hy^{-1}\cap Vxy^{-1}}$. Since $H$ is a subgroup and $y\in H$, $Hy^{-1}=H$. Also, since $y\in Vx$, $xy^{-1}\in V^{-1}$ so $Vxy^{-1}\subseteq VV^{-1}\subseteq U$. Thus $$xy^{-1}\in\overline{H\cap U}\subseteq \overline{H\cap\overline{U}}=H\cap \overline{U}.$$ In particular, $xy^{-1}\in H$, so since $y\in H$ and $H$ is a subgroup, $x\in H$.

Answer 2

Let $(x_n, n\in I)$ be a net where $A$ is a directed set. Suppose $x=lim_nx_n, x_n\in H$, write $y_n=x^{-1}x_n$, $\lim_ny_n=e$, there exists $N$ such that $n>N$ implies that $y_n\in U$. Let $n_0>N$ $z_n=x_n^{-1}x_{n_0}$, $\lim_nz_n=x^{-1}x_{n_0}\in U$ implies that there exists $M$ such that $n>M$ implies that $x_n^{-1}x_{n_0}\in U\cap H\subset\bar U\cap H$ we deduce that, $x^{-1}x_{n_0}=\lim_{n\geq M}x_n^{-1}x_{n_0}\in \bar U\cap H$ since $\bar U\cap H$ is closed. This implies that $x^{-1}x_{n_0}\in H$ and $x\in H$.

https://en.wikipedia.org/wiki/Net_(mathematics)#Properties

Answer 3

First, a bit of point set topology: $\bar U \cap H$ closed is equivalent to $U \cap H$ is closed in $U$.

Second, by homogeneity, for every $h\in H$, there exists $U_h$ neighborhood of $h$ so that $U_h \cap H$ is closed in $U_h$ ( take $U_h = h\cdot U$)

The above means that $H$ is locally closed in $G$. This is equivalent to : $H$ is the intersection between a closed set and an open set, or, what we are really after: $H$ is open in its closure $\bar H$.

Now, $\bar H$ is a topological group ( with the induced topology. Let's now recall: an open subgroup of a topological group is also closed. Indeed, any coset will be open, so any union of cosets. Therefore $H$ is also closed in $\bar H$, and so equal to $\bar H$, closed.

Take home message: a locally closed subgroup is closed.