Hereditarily compact space vs a space in which every subset is compact

Question

I read the definition of a hereditarily compact space in topospaces.subwiki.org, it says:

A topological space is termed hereditarily compact if every subset of it is compact in the subspace topology.

I don't understand how this is any different from saying: every subset is compact. Can someone explain the difference?

Answer 1

There is no difference. Note that compactness is usually defined as a property of topological spaces, not as a property of subsets of topological spaces. So when we say a subset of a space is compact, that really is a shorthand for saying the subset is a compact topological space when you give it the subspace topology.