Relative homology groups

Question

The local homology of a manifold $X$ at a point $x$ is defined as the relative homology $H_n(X, X-{x};\ \mathbb Z)$. It holds true for relative homology that under certain conditions $H(X,A) = H(X/A)$. For example it is satisfied for connected smooth surfaces like a torus. But doesn't it mean that $X/(X - \{x\})= x$? And if so, the $H_n(X, X-{x})$ is just the homology of a point? Please help me figure out possible mistakes in my geometric thinking.

Answer 1

The conditions you mention in order for $H_n(X,A)=\tilde{H}_n(X/A)$ is that the closure of $A$ has a neighbourhood which deformation retracts on to $A$. This is not possible in the case $A=X\setminus\{x\}$ for obvious reasons. Therefore the condition is not satisfied. Also, the space $X/(X\setminus\{x\})$ is a topological space with two points $\{x\}$ and $X\setminus\{x\}$, and (assuming $X$ is a manifold) has the homeomorphism type of the Sierpiński space.