Thread: retract in a topological space

If X is a topological space, what would be an example of a retract of X which is not closed?

I know that if X is Hausdorff, every retract of X must be closed. So I think we need to come up with a topological space X which is not Hausdorff.

[A C X is a retract of X if there exists a continuous map r:X-->A such that r(y)=y for each y in A].

Originally Posted by math8

If X is a topological space, what would be an example of a retract of X which is not closed?

I know that if X is Hausdorff, every retract of X must be closed. So I think we need to come up with a topological space X which is not Hausdorff.

[A C X is a retract of X if there exists a continuous map r:X-->A such that r(y)=y for each y in A].

Let T be a finite complement topology on . We know that is not closed in (2 is not finite in ).

Let r be a function such that r(2i-1)=r(2i)=2i for each . r is well-defined. For any closed subset A of , is closed in . Thus r is continuous. We also have r(2i)=2i for each i in .

Thus, 2 is a retract of , which is not closed in .