TITLE: Removing a single point alters homology
QUESTION [6 upvotes]: Let $X$ be a space whose homology groups are finitely generated. In order to avoid trivial cases, suppose that $X$ is not a singleton. Must there exist a point $p \in X$ such that $X\setminus\{p\}$ does not have the same homology groups as $X$? This seems implausible, but I have been unable to think of a counterexample. It seems to hold for most of the "usual" spaces one considers, eg. spheres, tori, $\mathbb R^n$, and various products and wedge sums thereof. Any ideas?

REPLY [3 votes]: If $M$ is a manifold, then let $p\in M$ be the point you want to remove, and $A$ a Euclidean neighborhood of that point.  Then we have by excision
$$ H_\ast(M)\cong H_\ast(M,A)\cong H_\ast(M-\lbrace p\rbrace,A-\lbrace p\rbrace).$$
Since $A-\lbrace p\rbrace$ is homotopic to $S^{n-1}$, the LES on relative homology shows $H_{n}(M)\cong H_{n}(M-\lbrace p\rbrace)\oplus\mathbb{Z}$.