# Discussion of local homology groups on Hatcher's Algebraic Topology

In p. 126 of Hatcher's Algebraic Topology, there is a discussion after theorem 2.26 about local homology groups. In particular, he says that the local homology groups of $X$ at a point $x \in X$ are the groups $H_n(X, X - \{x \})$. Until here everything is clear.

But then he uses excision theorem to claim that for every open neighborhood $U$ of $x$, $H_n(X, X - \{x\}) \equiv H_n(U, U- \{x \})$. What I don't understand is why he is not assuming that $x$ is a closed point of $X$. If it were closed, then we just use excision theorem with the open sets $U$ and $X - \{x\}$, but without this assumption I don't understand how can he use the excision theorem.

So the question is: do we need to assume that $x$ is closed in $X$?

Thank you!

• In my version of that book, in the last paragraph on page 126, he writes "...assuming points are closed in $X$, ..." – Stefan Hamcke Feb 25 '15 at 15:47
• Actually this is specified also in the pdf linked in the question... – Dario Feb 25 '15 at 16:06
• Thank you! In my printed version it is not mentioned, I should have checked more carefully the online version :) – Pedro A. Castillejo Feb 25 '15 at 16:41