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