Subject: Re: three postings
From: Allen Hatcher
Date: Sat, 24 Mar 2007 09:47:56 -0400
Here is an answer to the first part of the question about numerable
contractible spaces. An example of such a space that is not homotopy
equivalent to a CW complex is the (unreduced) suspension SY of the
subspace Y of the real line consisting of the sequence 1/2, 1/3, 1/4, ...
together with its limit 0. The fact that SY is not homotopy equivalent
to a CW complex follows from a nice little exercise: A path-connected
compact space X with pi_1(X) not finitely generated cannot be homotopy
equivalent to a CW complex. [Hints: (1) The continuous image of a
compact space is compact. (2) A compact subspace of a CW complex is
contained in a finite subcomplex. (3) A finite CW complex has pi_1
finitely generated.]
Allen Hatcher
> Subject: topology
> From: e.schwamb@t-online.de (Eugenia Schwamberger)
> Date: 22 Mar 2007 22:03 GMT
>
> Hello,
>
> I am writing a master thesis on numerable contractible spaces, i.e.
spaces X which have a numerable cover {Ua} such that the inclusions Ua
--> X are nullhomotopic. Each space of the homotopy type of a CW-complex
is numerable contractible. Does anyone know of an example of a numerable
contractible space which is not of the homotopy type of a CW-complex?
>
> Is the function space Top(K, X) numerable contractible if X is
numerable contractible and K is compact?
>
> Thank you,
> Eugenia Schwamberger
>