Date: Dec 4, 2012 1:27 AM
Author: Ali  Taghavi
Subject: Compactification of  space  without changing the  homotopy type

Let  X  be  a locally  compact  Hausdorff space. Is  there a  compact  Hausdorff  space Y which  contains  X as  a  dense subspace and  is  homotopic  equivalent to X.
In the other word, is there a  compactification of X with the same  homotopy type as  X.
The motivation: Let X  be the open unit ball in R^n,then one  point compactification of X, S^n, is  not homotopic  equivalent to X  but the closed unit ball is a compactification of X  with the same  homotopy type.
A translation of this  question in  the world of  (noncommutative) C* algebras  is the following:
Let  A  be  a  C*  algebra. Does there exist a  unitization of  A  wich is  homotopic equivalent to A
Thanks