Are the rational numbers Q, with the relative topology as a subset of the real numbers, locally compact ? why?
You wrote locally path connected, when I assume you meant compact like you said in the actualy body.
The answer is no. Let be any neighborhood of in . Then, . Also, remember then that if are top. spaces and is a superspace of then will be a compact subspace of if and only if it's a compact subspace of and thus for to be compact in we need that is compact in . But, evidently this is not compact in , why?