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