The wikipedia page on Borel-Moore homology claims to give several definitions of it,
all of which are supposed to coincide for those spaces $X$ which are homotopy equivalent to a finite CW complex and which admit a sufficiently nice embedding into a smooth manifold.
My question concerns one of the definitions on that page: it is asserted that if $X$ is a space and $\bar X$ is some compactification, then there is a definition of Borel-Moore homology given by
$$
H_*^{\text{BM}}(X) := H_*(\bar X,\bar X \setminus X) .
$$
The entry additionally requires that the pair $(\bar X,X)$ should be a CW pair for this to work.
It then goes on to assert that $\bar X = X^+$, the one-point compactification, will suffice for this.
Frankly, I don't see how $(X^+,X)$ will form a CW pair in most instances. For example suppose $X = \Bbb Z$ is the set of integers which is considered as a discrete space. Then in this instance $(X^+,X)$ certainly fails to be a CW pair.
Another example: $X = S^1 \setminus \ast$ with one point compactification $S^1$. Then
$(S^1,S^1 \setminus \ast)$ is not a CW pair either.
My Questions:
(1) For what class of spaces $X$ does
$H_*(X^+,+)$ coincide with definition of Borel-Moore homology given by locally finite chains?
(2) Does the wikipedia entry contain a mistaken assumption? Perhaps we do not need to assume that $(X^+,X)$ can be given the structure of a CW pair?
Note: if $X = \Bbb Z$ and if we use the above as a definition of Borel-Moore homology then
$
H_0^{\text{BM}}(X)
$
is a free abelian group whose generators are given by the underlying set of $\pi_0(X) = \Bbb Z$. This is clearly the wrong answer: it should be the countably infinite cartesian product of copies of the integers indexed over $X$ instead (using, say, the definition of Borel-Moore homology given by locally finite chains).
Another Question:
(3) Is there a definition the above kind (using compactifications)
which will work (i.e., coincide with the locally finite chain definition)
for $X = \Bbb Z$?
(I suspect not, since ordinary singular homology in degree zero is always free abelian.)
Incidentally, later in the page it lists the main variance property: Borel-Moore homology is supposed to be covariant with respect to proper maps. The page gives a
proof using the above definition. But since the above definition doesn't work in general, I don't see how this is really supposed to be a proof.
Best Answer
In Chriss & Ginzburg, it's asserted that the Borel-Moore homology is given by $H_*(\bar X, \bar X\setminus X)$ if $\bar X$ is the one-point compactification or any compactification such that the inclusion is cellular. They very pointedly do not assume that the inclusion of $X$ into $X^+$ is cellular. They reference Bredon but don't cite a particular chapter.