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The table on the separation axioms page looks very messed up. If I open the table directly at main separation axioms – table it looks fine but on the separation axioms page, all of the links are gone. How do we fix it?
I re-rendered (made a trivial edit to) the table page, the other page looks fine now I think. This was probably a relic of the teething issues with the change in renderer last year.
Great! thanks
I added to the introduction a couple of remarks mentioning separation axioms in terms of lifting properties with respect to finite topological spaces (i.e. preorders).
Also, I noted that the page mentioned but did not define Axiom T5, so I made a change making it clear it means “completely normal” (=heriditarily normal).
Anonymous
Where the paragraph ends with:
… i.e. the function determined by two distinct points is not continuous as a map from the indiscrete space.
I feel that a conclusion is missing that explicitly explains the promised lifting.
Not having looked at how this works, I gather the sentence could be completed as:
…or concersely: Every continuous such function necessarily factors through the point space.
But that would be an “extension property”, not a lifting property. (?)
(I understand that I will find this explained in more detail in the entry on separation by lifting, and I’ll look at that when I have more leisure. But just to mention that the idea section here could make this clearer.)
Where the paragraph ends with:
… i.e. the function determined by two distinct points is not continuous as a map from the indiscrete space.
I feel that a conclusion is missing that explicitly explains the promised lifting.
I added such a conclusion.
…or concersely: Every continuous such function necessarily factors through the point space.
But that would be an “extension property”, not a lifting property. (?)
Correct. But an extension property is a form of a lifting property. Now I write the lifting properties for T0, T1, and T2 explicitly, although in a rather verbose manner.
(I understand that I will find this explained in more detail in the entry on separation by lifting, and I’ll look at that when I have more leisure. But just to mention that the idea section here could make this clearer.)
The chief difficulty there is introducing notation for maps of preorders (equivalently, maps of finite topological spaces)….
Anonymous
I included diagrams for separation axioms T0-T4 as lifting properties with respect to morphisms of finite topological settings. One should probably fix the typesetting and put diagrams next to each other, not below each other. I hope the diagrams are self-explanatory, with remarks provided.
Anonymous
There’s a spelling error in the section “Beyond the classical theory”:
The space is $T_1$ if and only if the sepcification order is the equality relation.
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