In a locally compact hausdorff space X for every x in X there exists a compact set that contains an open set V s.t. x is in V. I have to prove this is equivalent to for every open set U s.t. x is in U, there exists a open set V s.t x is in V and closure of V is compact and contained in U. I know how to prove it using the one point compactification of the space X, is there an easier way to prove this. for the reverse implication is easy. any hints would be helpful