what is underlying topological space?

Question

what is underlying topological space? I don't get the definition of this to topology book. I wonder the definition of this topological space.

Answer 1

If you're dealing with a topological {group, vector space, ring or any other algebraic structure}, referring to "the underlying topological space" usually means to look at the object only as a topological space.

You can find this with Lie groups, the real spaces $\,\Bbb R^n\,$, general Hilbert or Banach spaces, etc.