Subject: Re: three postings
From: Allen Hatcher <hatcher@math.cornell.edu>
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
>