In Spanier's Algebraic Topology, he defined CW complexes assumed an additional strange condition: the cell must have the coherent topology with the characteristic map and the inclusion map of its boundary. What is the meaning of this condition? Some other author seems never use it, and they asked the space to be Hausdorff. Are the CW complexes Hausdorff in Spanier's way? Do the various definitions agree? Thanks!

3$\begingroup$ math.stackexchange.com would be a good website for your question. I think it would be a good idea to put a little more effort into it and list the definitions you're referring to. People likely do not have the textbooks you're referring to on hand. $\endgroup$ – Ryan Budney Mar 21 '12 at 9:27

$\begingroup$ I thought this had been answered in mathoverflow.net/questions/74863/… $\endgroup$ – Ronnie Brown Mar 25 '12 at 10:04
It basically says that a CW complex has the coherent topology from its closed cells. This should be wrapped up into any definition of a CW complex that you see.
For example in Massey's book it is equivalent to statement (iv):
A subset $A$ is closed if and only if $A \cap \bar{e}$ is closed for all $n$cells, $e^n,n=0,1,2,\ldots$
(You can see this equivalence at the wikipedia page).