Re: Definitions of T_0, T_1, etc. Topological Spaces

On Dec 7, 4:54 pm, Jonathan Groves <JGro...@xxxxxxxxxx> wrote:
Dear All,

A friend of mine loaned me his book "Counterexamples in Topology" by Lynn Arthur Steen and J. Arthur Seebach, Jr. Their definitions of T_3, T_3.5, and T_4 do not assume the spaces are T_1. But their definitions of completely regular, regular, and normal assume the spaces are T_1. They also mention the terminology T_3 and regular are used the way the authors use it but that others use these two terms to mean the same thing (Munkres, for example), which I suppose assumes T_1 but they're not explicit about whether these kinds of authors assume T_1 or not. And still others reverse the authors' meanings of T_3 and regular (meaning that regular does not assume T_1 but that T_3 does assume T_1). Same comments for T_3.5 and T_4.

This book does mention T_2.5 spaces, and these are also called completely Hausdorff. They also say that T_0 spaces are sometimes called Kolmogorov spaces and that T_1 spaces are also called Frechet spaces.

How do you pronounce "Frechet"? (There's an accent over the first "e," but I cannot write it on the keyboard.) And has anyone ever heard of these terms I mentioned in the paragraph just before this one? I'm sure someone here has, but I'm just wondering how common these terms are.

And I'm still confused about what is the most common usage of the terms T_3, T_3.5, T_4, regular, completely regular, and normal. Maybe no one knows, but I'm curious anyway.

By the way, I know there are T_5 and T_6 spaces (maybe also T_i for i's bigger than 6), but I'm not as familar with these terms. Our topology course doesn't go into these spaces.

i've seen T0 spaces called kolmogorov quite frequently
but T1 frechet less commonly (though it is not rare)

on reason for assuming the implicative forms of the separations axioms
ie. "higher" separation axioms imply lower
T5 -> T4 -> T3 -> T2 -> T1 -> T0
is because they are the ones that often generalise nicely
and having the hierarchy makes proving properties less work

one such generalisation
for instance
is http://www.tbi.univie.ac.at/papers/Abstracts/01-pfs-017.pdf
(the chart on page 10 shows the implicative structure)

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galathaea: prankster, fablist, magician, liar
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