Let XT be a topological spaceA subset X Suppose that for all

Let (X,T) be a topological space,.A subset X. Suppose that for all x epsilon A, U epsilon T s.t. x epsilon U subset A. Show that.4 is open in X.

Solution

take the union of all such open sets

i.e. Union of U_x for all x in A

Union U_x is open since U_x is open for all x.

and Union U_x = A (Union U_x is a subset of A and A is subset of Union U_x)

So A is open