What different topological properties do spaces of different separation axioms have?
I am looking for somehow "categorizing" topological (and non-topological) properties of spaces according to their separation axioms. (e.g. how functions behave, what properties may coincide...)
I have never seen any complex view on this topic.
What I have in mind is something like Venn diagrams from the most general separation axiom to the most "strict" ... And for each $T_i$ the properties that hold for spaces satisfying that axiom.
For example, for Tychonoff spaces, one of the properties would be that they admit Hausdorff compactifications etc.
Could you recommend any resource where to find such view? Or can you think of any properties specific for the individual separation axioms?
Thank you.
One book that may be of interest is Counterexamples in topology. It gives a long lists of topological spaces with their properties. This would provide you with lots of counterexamples to theorems when the separation axiom required is relaxed a bit (as in "[Theorems] holds for $T_2$ spaces but [Space] gives a $T_1$ counterexample")