A set A subset of X in a topological space X is said to be d

A set A subset of X in a topological space X is said to be dense in X if A bar = X. Show that if U is open then U subset of (A intersection U) bar

Solution

Since ?A ? {V ? A : V ? X} = {V ? A ? Y : V ? X} = {U ? A : U ?o Y }, the relative topology on A inherited from X is the same as the relative topology on A inherited from Y . Since connectivity is a statement about the relative topologies on A, A is connected in X iff A is connected in Y