There are exactly eight types of connected proper subsets of the reals:
One of those is compact. As a space is itself not compact.
me and my friend got into a little bit of argument...
the question asks to find all subsets of the real numbers that are both compact and path connected. i was considering the closed interval [a,b] where a,b are in the real numbers, but my friend was telling me it should be the set of real numbers, which is a subset of itself. but i know that for the subset to be compact it must be closed and bounded by the heine-borel theorem... so is [a,b] all of the subsets or are there more? thanks in advance!