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

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You’ll have to give more context: you haven’t given enough information for the question to be answered. – Brian M. Scott Oct 24 '12 at 12:59

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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.

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