- #1

nightingale123

- 25

- 2

## Homework Statement

Let ##X=([1,\infty)\times\{0\})\cup(\cup_{n=1}^{\infty}\{n\}\times[0,1])## and ##Y=((0,\infty)\times\{0\})\cup(\cup_{n=1}^{\infty}\{n\}\times[0,1])##

##a)##Find subspaces of of the euclidean plane ##\mathbb{R}^2## which are homeomorphic to the compactification with one point ##X^+## and ##Y^+##

##b)## are any of the subspaces ##X## and ##Y## retracts of ##\mathbb{R}^2##

## Homework Equations

## The Attempt at a Solution

I was able to find the required subspaces for ##a)## and proved that they are homeomorphic, however I'm having some problems with ##b)##. I know that ##Y## cannot be a retract as it is not closed.

I got that since ##\mathbb{R}^2## is a metric space therefore ##T_{2}## and we know that if ##f,g:X\to Y## are continuous and ##Y\in T_{2}## then the set ##\{x|f(x)=g(x)\}\subset Y## is closed. However in this case ##Y=\mathbb{R}^2## and if there exists a retraction ##r:\mathbb{R}^2\to Y\subset \mathbb{R^2} ## the set ##\{x|r(x)=id_{x}(x)=x\}##,which is ##X##, must be closed. And since it is not. ## X ## cannot be a retraction. Can someone correct me here if I did something wrong?

However this argument does not work for ##X## as ##X## is closed. ##X## is also connected so that fails too. I can't find a homeomorphism between ##X## and ##\mathbb{R}##. Could someone clarify how we could determine whether X is a retract or not?

EDIT: I'm sorry for posting this in the advanced physics section. I just noticed it and I don't know how to change it

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