Question

Reading about separation axioms, I wonder: Is there a separation axiom weaker than $T_2$ but stronger than $T_1$? $T_{1.5}$? I suppose there are some separation axioms stronger that $T_6$, how many are there?

Answer 1

There is a separation axiom between $T_1$ and $T_2$ that Aull (Separation of bicompact sets, Math. Annalen 158 (1965), 197–202) calls $J_1$ and that Mukherji (On weak Hausdorff spaces, Bull. Calcutta Math. Soc. 58 (1966), 153–157) calls $T_2'$; namely, "every compact subspace is closed." Such spaces are sometimes called "weak Hausdorff spaces," although this term is sometimes used to mean something else. See Hoffmann (On weak Hausdorff spaces, Arch. Math. (Basel) 32 (1979), 487–504) for further discussion of other separation axioms between $T_1$ and $T_2$.

Answer 2

I've looked into this before, and I never heard of $T_{1.5}$. However, there are notions of $R_0$ and $R_1$ which can combine to get the usual $T$-separation axioms. For instance, a space is $T_1$ iff it is $R_0$ and $T_0$. A space is $T_2$ iff both $T_0$ and $R_1$. This might give you some ideas for how to create a $T_{1.5}$ if you wanted to. A good reference is the wikipedia article on the separation axioms.

Another notion below $T_2$ but above $T_0$ is that of a sober space. Perhaps this will work for whatever application you have in mind.

As for "above" $T_6$, I think at that level you are basically at being a metric space. Recall that a metric space satisfies all the separation axioms, and recall the Metrization Theorems about how close the various separation axioms are to implying metrizability. Every discrete topology is metrizable, so I'd put that past metrizable on the scale, as perhaps the ``nicest'' space possible.