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correction of typos
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user43326
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I think I should be posting this as a comment, but as i don't have enough reputation to comment I take the liberty to post this as an answer.

I am here just elaborating on Eric Wofsey's answer.

Let's take the case, say, $X=\mathbb{N}$ and $Y=I=[0,1]$. Then $Map(X,Y)$ is Hilbert cube.

On the other hand, <a http://en.wikipedia.org/wiki/CW_complex#Inductive_definition_of_CW-complexes> the definition of a CW-complex the definition of a CW-complex says that a CW complex has to be the union of finite skelta, and for reasons explained in Eric Wofsey's answer, this is not the case.

I think I should be posting this as a comment, but as i don't have enough reputation to comment I take the liberty to post this as an answer.

I am here just elaborating on Eric Wofsey's answer.

Let's take the case, say, $X=\mathbb{N}$ and $Y=I=[0,1]$. Then $Map(X,Y)$ is Hilbert cube.

On the other hand, <a http://en.wikipedia.org/wiki/CW_complex#Inductive_definition_of_CW-complexes> the definition of a CW-complex says that a CW complex has to be the union of finite skelta, and for reasons explained in Eric Wofsey's answer, this is not the case.

I think I should be posting this as a comment, but as i don't have enough reputation to comment I take the liberty to post this as an answer.

I am here just elaborating on Eric Wofsey's answer.

Let's take the case, say, $X=\mathbb{N}$ and $Y=I=[0,1]$. Then $Map(X,Y)$ is Hilbert cube.

On the other hand, the definition of a CW-complex says that a CW complex has to be the union of finite skelta, and for reasons explained in Eric Wofsey's answer, this is not the case.

Source Link
user43326
  • 3.1k
  • 17
  • 25

I think I should be posting this as a comment, but as i don't have enough reputation to comment I take the liberty to post this as an answer.

I am here just elaborating on Eric Wofsey's answer.

Let's take the case, say, $X=\mathbb{N}$ and $Y=I=[0,1]$. Then $Map(X,Y)$ is Hilbert cube.

On the other hand, <a http://en.wikipedia.org/wiki/CW_complex#Inductive_definition_of_CW-complexes> the definition of a CW-complex says that a CW complex has to be the union of finite skelta, and for reasons explained in Eric Wofsey's answer, this is not the case.