# Retract condition?

• May 17th 2011, 10:58 PM
aharonidan
Retract condition?
Let $X$ be a Topological space.
a . show that for a point $a \in X$ there is a retract $f : X \rightarrow {a}$
b . if $A \subset X$ is finite, and $X$ is Hausdorff with a finite amount of connected components, what is the Necessary and sufficient condition for a retract $f : X \rightarrow A$ ?

thanks!!
• May 18th 2011, 12:36 AM
Tinyboss
The definition of a retract $f:X\to \{a\}\subset X$ 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.
• May 18th 2011, 12:49 AM
aharonidan
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
• May 18th 2011, 04:42 AM
Tinyboss
Think about how f would act on a path connecting distinct points of A.