1. The problem statement, all variables and given/known data Let X be a topological space, a subset S of X is said to be locally closed if S is the intersection of an open set and a closed set, i.e S= O intersection C where O is an open set in X and C is a closed set in X Prove that if M,N are locally closed subsets then M union N is locally closed. 3. The attempt at a solution So M = O intersection C and N = O' intersection C' where O, O' are open sets in X and C, C' are closed sets in X. It follows that M union N = (O intersection C) U (0' intersection C'). From h ere I played with this expression a while using distributive laws but got stuck, somewhere I end up with the union of a closed set and an open set, i.e O U C, but I think we cannot said anything about this particular union. Maybe I'm missing some useful set-theoretical identity. Can you please help?