This question was asked in my Topology quiz now over and I was unable to solve it.
Let (X,T) be a $T_1$ space and let (Y,U) be a topological space and let f be a closed map of X onto Y. Prove that (Y,U) is a $T_1$ - space.
Attempt: Let x' and y' be distinct elements in Y. Then there exists x,y in X such that f(x)=x' and f(y)=y' and x$\neq$ y. So, there exists open sets U and V such that x$\in $U and y $\in V$ such that x doesn't lies in V and y doesn't lies in U.
But I am not able to construct open sets in Y which will satisfy $T_1$ property.
Can you please help.