TITLE: Is $Alt_\omega$ a dense subgroup of a non-discrete locally compact topological group?
QUESTION [12 upvotes]: Let $S_\omega$ be the group of bijections of the countable ordinal $\omega:=\{0,1,2,\dots\}$ and $Alt_\omega$ be the subgroup of $S_\omega$ consisting of even permutations of $\omega$ (i.e., the compositions of even number of transpositions). By Onofri-Schreier-Ulam Theorem, $Alt_\omega$ is the smallest non-trivial normal subgroup in the permutation group $S_\omega$.
It is known that
$\bullet$ the group $Alt_\omega$ is not isomorphic to a subgroup of a compact topological group;
$\bullet$ the group $S_\omega$ is not isomorphic to a dense subgroup of a non-discrete locally compact topological groups.
These two results motivate the following 

Problem 1. Is the group $Alt_\omega$ isomorphic to a dense subgroup of a non-discrete locally compact topological group?

Since the group $Alt_\omega$ is simple, this problem can be reformulated in the following form.

Problem 2. Let $h:Alt_\omega\to G$ be a homomorphism of $Alt_\omega$ to a locally compact topological group $G$. Is the image $h(Alt_\omega)$ discerete?

Remark. By the answer of @YCor to this question, 
for every homomorphism $h:S_\omega\to G$ to a locally compact topological group $G$ the image $h(S_\omega)$ is discrete.

REPLY [4 votes]: Corollary 1.5 of this Vissarion Belyaev's paper says that

A group containing an infinite inert residually finite subgroup
  embeds into a non-discrete locally compact group.

Here, a subgroup $H$ of a group $G$ is called inert if $H\cap g^{-1}Hg$ has a finite index in $H$ for any $g\in G$.
So, in the alternating group $Alt_\omega$, for instance, the (abelian) subgroup generated by the 3-cycles $(1\;2\;3),(4\;5\;6),(7\;8\;9),\dots$
is inert. infinite, and residually finite .   
Thus, the answer to Question 1 is yes (and to Question 2 is no).