Let be a Topological space.

a . show that for a point there is a retract

b . if is finite, and is Hausdorff with a finite amount of connected components, what is the Necessary and sufficient condition for a retract ?

thanks!!

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- May 17th 2011, 10:58 PMaharonidanRetract condition?
Let be a Topological space.

a . show that for a point there is a retract

b . if is finite, and is Hausdorff with a finite amount of connected components, what is the Necessary and sufficient condition for a retract ?

thanks!! - May 18th 2011, 12:36 AMTinyboss
The definition of a retract 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 AMaharonidan
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 AMTinyboss
Think about how f would act on a path connecting distinct points of A.