Separation Axiom Visualisations

A couple days a go, I saw some nice visualisations of separation axioms on Wikipedia. Unfortunately, it wasn’t a full set. Well, here is a full set (well, T0, T1, T2, T2 1/2, T3, T4, T5):

T0, Kolmorgov:

T1, Frechet:

T2, Hausdorff:

T2 1/2, Completely Hausdorff:

T3, Regular:

T4l:

T5:

(I’ll put them on WIkipedia at some point… Tomorrow, maybe.)

T3 1/2 isn’t in here because of the difficulties in visualising it. The definitions of the separation axioms I used to make these are taken from Counterexamples In Topology.

Update:

The following visualizations of T3.5 and T4.5 are based off what I saw in a topology course I’m taking with Dror Bar-Natan. The course is based on Munkres, and the same visualisations may be there.

T3.5:

Visualization of teh T3.5 separation axiom

(the red is the real valued function.)

T4.5:

The T4.5 Separation Axiom

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3 Responses to “Separation Axiom Visualisations”

  1. Sniffnoy Says:

    These are mostly good, but T5 looks essentially the same as T4. (I’m not sure T5 is worth bothering with, really.)

    What is “T4.5″? Is that supposed to mean “Every two closed sets are separated by a function”? You’re aware that’s true in every normal space, right? (Indeed, you could actually make multiple versions of some of these, showing definitions and consequences. Like, T4 as the definition, and use your T4.5 picture as the consequence. Or your T3 picture as the definition, and a variant of your T2.5 picture as the consequence (since in any regular space, a point and a closed set can always be separated by closed neighborhoods).)

    Actually all your function based ones are kind of cluttered; I’d suggest removing the open sets from them and let the function stand on its own. Although right now it’s not really too clear what’s going on with the function, without your explanation. Here’s perhaps a better idea: Pick two colors (let’s say red and white for the sake of example), one for your one closed set and one for the other, actually compute a separating function (since they’re closed sets in the plane), and use this to make a gradient. So your one set would be red, the other white, and the rest of the picture varying shades of pink depending on how close it was to each of them.

    …although that might look too much like “precisely separated by a function” rather than just “separated by a function”, come to think of it, since, well, it would be precisely separating them with a function. I guess you could have the sets in black, inside the pure red/white region? And then for precise separation do the same thing except there is no pure red/white region because those regions are both all black due to being the sets?

    • christopherolah Says:

      > These are mostly good, but T5 looks essentially the same as T4. (I’m not sure T5 is worth bothering with, really.)

      This is probably true :)

      > What is “T4.5″? Is that supposed to mean “Every two closed sets are separated by a function”? You’re aware that’s true in every normal space, right? …

      I know Uryhsohn’s Lemma of course. My understanding is that T4½ historically existed before he proved they were the same as T4 ones…

      >I’d suggest removing the open sets from them and let the function stand on its own…

      That would be nice. I threw these together quite quickly in Inkscape (which doesn’t really support that) which doesn’t have nice gradients, which is the main reason these a bit sucky.

      • sniffnoy Says:

        > That would be nice. I threw these together quite quickly in Inkscape (which doesn’t really support that) which doesn’t have nice gradients, which is the main reason these a bit sucky.

        Hm, troublesome. I don’t know what you would use for that.

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