How to best "sort" Tychonoff spaces, so it is mutually exclusive, collectively exhaustive?
Why am I asking: When describing certain mathematical phenomenon in a paper, it may be useful to go through all kinds of spaces and study different kinds of spaces behave.
My question: Is there any "standard" classification of topological spaces into "categories"? I can only think of having one criteria and sorting spaces into two mutually exclusive groups (e.g. EUCLIDEAN and NON-EUCLIDEAN... or DISCRETE and CONTINUOUS... or FINITE and INFINITE.)
However, I am looking into something less aggregate, some system which would provide reasonable number of groups for the spaces and in each group there would be few examples.
My ideas: I thought maybe sorting the spaces according to separation axioms would be good idea? (E.g. higher separation axiom, more conditions on the space?) Or sorting spaces according to the most common topology on it? (However, this would make overlapping groups of spaces... And there are too many topologies.)
(I concentrate on Tychonoff spaces (completely regular & Hausdorff), but classifying more general types of spaces can be interesting too.)