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0)The solution by Gleason, Montgomery, and Zippin of the fifth Hilbert's problem says that any òpcally euclidean topological group admits a compatible analytical structure.

1)By Cartan--Von Neumann theorem on closed subgroups, if $G$ is a Lie group (i.e. smooth group manifold) and $H$ is a closed subgroup of $G$, then $H$ is an embedded Lie subgroup of $G$, so its connected components w.r.t. the subspace topology and its connected components w.r.t. this Lie group structure are the same.

2)The connected components of a topological manifold are open.

Now 0),1) and 2) should be conclusive for more than your question, i.e.:

If $G$ is a locally euclidean topological group and $H$ is a totally disconnected, closed subgroup of $G$ then $H$ is discrete topological subgroup of $G$.

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0)The solution by Gleason, Montgomery, and Zippin of the fifth Hilbert's problem says that any topological group admits a compatible analytical structure.

1)By Cartan--Von Neumann theorem on closed subgroups, if $G$ is a Lie group (i.e. smooth group manifold) and $H$ is a closed subgroup of $G$, then $H$ is an embedded Lie subgroup of $G$, so its connected components w.r.t. the subspace topology and its connected components w.r.t. this Lie group structure are the same.

2)The connected components of a topological manifold are open.

Now 0),1) and 2) should be conclusive for more than your question, i.e.:

If $G$ is a locally euclidean topological group and $H$ is a totally disconnected, closed subgroup of $G$ then $H$ is discrete topological subgroup of $G$.

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0)The solution by Gleason, Montgomery, and Zippin of the fifth Hilbert's problem says that any topological group admits a compatible analytical structure.

1)By Cartan--Von Neumann theorem on closed subgroups, if $G$ is a Lie group (i.e. smooth group manifold) and $H$ is a closed subgroup of $G$, then $H$ is an embedded Lie subgroup of $G$, so 1)its its connected components w.r.t. the subspace topology and its connected components w.r.t. this Lie group structure are the same.

2)The connected components of a topological manifold are open.

Now 10),1) and 2) should be conclusive for more than your question, if I am not wrong.i.e.:

If $G$ is a topological group and $H$ is a totally disconnected, closed subgroup of $G$ then $H$ is discrete topological subgroup of $G$.

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