# Thread: Retract condition?

1. ## Retract condition?

Let $\displaystyle X$ be a Topological space.
a . show that for a point $\displaystyle a \in X$ there is a retract $\displaystyle f : X \rightarrow {a}$
b . if $\displaystyle A \subset X$ is finite, and $\displaystyle X$ is Hausdorff with a finite amount of connected components, what is the Necessary and sufficient condition for a retract $\displaystyle f : X \rightarrow A$ ?

thanks!!

2. The definition of a retract $\displaystyle f:X\to \{a\}\subset X$ is a continuous function from X to {a} such that f(a)=a. And that pretty much nails it down for you.

For (b), assuming you mean "finite" when you write "final", think about where the points of A are distributed within the components of X.

3. I corrected my spelling mistake.
I still don't get under what condition I can find a retract. Isn't the fact that both A and the connected components of X are finite is enough? I'm probably missing something...
do all points of A need to be in the same component of X?
thanks

4. Think about how f would act on a path connecting distinct points of A.