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.



On the wikipedia entry for Borel-Moore homology
In [Chriss & Ginzburg](http://books.google.com/books?id=lwS59rR78eIC&dq=chriss+and+ginzburg&hl=en&sa=X&ei=mkH3UbavFZal4AP1m4DgAw&ved=0CDEQ6AEwAA), 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](http://books.google.com/books?id=zGdqWepiT1QC&printsec=frontcover&dq=bredon+sheaf+theory&hl=en&sa=X&ei=gkP3UcPnOKT-4APamoHQBQ&ved=0CDoQ6AEwAA#v=onepage) but don't cite a particular chapter. 


