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!
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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):
(You can see this equivalence at the wikipedia page). 

