Basic Lie group Theory

Question

I have a couple of clarifying questions about Lie groups. (In particular matrix Lie groups)

1. when we say a closed subgroup of a Lie group G, do we literally mean closed in the usual topological sense of being closed (same with compact?)
2. when we say a lie subgroup of a Lie group G, do we mean a subgroup that has a manifold structure (versus a subgroup which wouldn't?)
3. path connected = connected in a matrix subgroup?

thank you!

Answer 1

1. Closed in the usual (topological) sense.

2. A subgroup which is also a submanifold.

3. Yes, these are the same because every manifold is locally path connected.