Having some difficult with general concepts of metric spaces: 1) What is the difference between a subset and a subspace. let's say we have metric space X. and A is a set in that space. Is A necessarily a metric space itself? 2) Why is the metric of X ( d(x,y) for x,y belonging to X ) necessarily finite? Isn't the set of all real numbers a metric space, then how can you say that distance between any two numbers is finite? Thanx!