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.

Ends of Complexes(maths.ed.ac.uk/~aar/books/ends.pdf). $\endgroup$6more comments