How do separation axioms carry over to subspaces? Some are clear -- it's easy to see that if any two points of a space X are separated by neighborhoods, then the same must be true of any subset S of X. But what about the nicer ones? Is it true that if S is a subset of a normal space, that S is itself normal? This one is less obvious... one example that worries me is this: Consider the union S of two open discs in R^2. (that aren't disjoint) Consider the two closed sets formed by restricting the boundaries of the two discs to S. We can't directly appeal to the normality of R^2, because the closure of these sets aren't disjoint in R^2. It is still easy to see S is normal, because it's homeomorphic to R^2, but that doesn't help me in the general case of a subset of an aribtrary normal space.