reduced homology

Overview

Reduced homology is a modification of singular homology so that the homology of a single point is always $0$, this makes some statements a little clearer and generalized axioms for homology to fit better.

Definition

For a topological space $X$ and $x_0\in X$, then the reduced homology of $X$ is

${\tilde{H}}_{\bullet }(X)=H_{\bullet }(X,x_0)$

where $H_{\bullet }(X,x_0)$ denotes the relative singular homology.

Note that for all $\bullet \ge 1$ then the reduced homology is the same as the usual homology. It can differ on $H_0$.