# How to find that the following set is connected and compact or not? [duplicate]

Let $$U$$ denote the set of all $$n\times n$$ matrices $$A$$ with complex entries such that $$A$$ is unitary. Then $$U$$, as a topological subspace of $$\mathbb{C}^{n^2}$$ is,

1. Compact, but not connected.
2. Connected, but not compact.
3. Connected and compact.
4. Neither connected nor compact.

I know what the terms are but I am really not getting the idea how to solve this. Please help.

• For the compactness, not that we are considering a problem in some finite dimensional vector space. So compactness is equivalent to bounded and closed. Also, since every norm on this space is equivalent, choose your favourite one. – G. Chiusole Apr 2 at 8:59
• I am sorry but I could not understand ''every norm on this space is equivalent'' and how does this specify the answer? – Huny Apr 2 at 9:00
• Hint: consider the application $f(U):=U^{*}U$. What can you tell us about $f$ and $f^{-1}(\text{Id})$? – Caffeine Apr 2 at 9:02