Topology connected space Prove that if Xtau is connected and

Prove that if, X_tau is connected and tau\' tau, then X_tau\' is also connected.

Solution

T\' is a subset of T and hence all points or paths that are contained in T\' are also contained in T.

And we are given that X_T is connected that means any two points x0 and x1 in X_T can be connected in

X_T via a continuous path. But T\' is a subset of T and hence any two points in X_T\' can also be connected

via a continuous path in X_{T\' .} Therefore , we can say that X_T\' is also connected.