Prove that X is connected if and only if the only subsets of

Prove that X is connected if and only if the only subsets of X that are simultaneously open and closed are X and .

Solution

(X, A ) is connected if and only if and X are the only subsets of X that are simultaneously open and closed in X.

Now we know that a topological space let say (X,A) is connected only when there is non-empty closed (open) subsets M and N of X such that

a) MN = ,

b) MN = X

In that case M = N^c and N = M^c and hence M and N are closed sets. Also X contains M ( ) i.e. a non-empty proper subset of X which is open and closed simultaneously in X.