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?
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2$\begingroup$ For instance you may consider "discrete", that we may call $T_7$ or maybe $T_\infty$ $\endgroup$– Pietro MajerCommented Jan 27, 2012 at 13:30
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$\begingroup$ For any space $X$ with more than one point one may consider the separation axiom "any two points can be distinguished by a continuous function to $X$." $\endgroup$– Qiaochu YuanCommented Jan 27, 2012 at 16:41
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$\begingroup$ I briefly considered the notion of separating arbitrary large disjoint subsets (where each set needed at least kappa elements). However, I can't distinguish this property from the space being discrete. Gerhard "Ask Me About System Design" Paseman, 2012.01.27 $\endgroup$– Gerhard PasemanCommented Jan 27, 2012 at 17:29
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2$\begingroup$ @yuan: that property is called completely Hausdorff, and is stronger than Hausdorff, not weaker. $\endgroup$– Henno BrandsmaCommented Jan 27, 2012 at 18:55
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2 Answers
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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 $T_2$.
answered Jan 27, 2012 at 16:25
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6$\begingroup$ Wilansky introduced KC (all compact sets are closed) and US (every convergent sequence has a unique limit) in the paper "between T1 and T2", American mathematical monthly, 74(1967), 261-264. We then have $T_2$ implies KC implies US implies $T_1$, and all implications cannot be reversed in general. $\endgroup$ Commented Jan 28, 2012 at 6:14
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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.
answered Jan 27, 2012 at 14:34