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lauha@lemmy.world ⁨1⁩ ⁨month⁩ ago

• someacnt@sh.itjust.works ⁨1⁩ ⁨month⁩ ago

  I am not a topologist, but I can try…

  A space (shape) is contractible if you can “contract” (shrink) it to a point without cutting, pinching or punching through holes. For example, a mattress is contractible, since you can shrink it to the center - just shrink along each line to the center. Meanwhile, a doughnut, a circle or a hollow sphere are not contractible, you can never remove the inner “hole” to shrink to a point without cutting.

  In general, any dimensional sphere is not contractible… Until it isn’t - infinite dimensional sphere is . Somehow, it loses the “hollow space” inside.

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  • zipsglacier@lemmy.world ⁨1⁩ ⁨month⁩ ago

    For each finite dimension n (1, or 2, or 3, etc…), the sphere in dimension n can’t be contracted because of that empty n-dimensional space it surrounds. But that same sphere is the “equator” of the sphere in the next higher dimension, n+1. There, the n-dimensional equator can contract along one of the hemispheres, to a pole. But then that whole (n+1)-dimensional sphere still isn’t contractible, because of the (n+1)-dimensional space it surrounds.

    BUT the (n+1)-dimensional sphere can contract along one of the hemispheres in the (n+2)-dimensional sphere. And so on.

    For any particular finite dimension n, there is an n-dimensional obstruction to contracting the sphere in that dimension. But if you go all the way to infinitely-many dimensions, there is no obstruction that ever stops contractibility of the infinite-dimensional sphere.

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  • ComicalMayhem@lemmy.world ⁨1⁩ ⁨month⁩ ago

    Sphere refers to the surface area, correct? Ball would be the volume inside?

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    • someacnt@sh.itjust.works ⁨1⁩ ⁨month⁩ ago

      Indeed.

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• dont@lemmy.world ⁨1⁩ ⁨month⁩ ago

  On what level? A proof? Or just the meaning of the words?

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  • lauha@lemmy.world ⁨1⁩ ⁨month⁩ ago

    What is contractible in mathematics? Wikipedia had article on contractible spaces but that was way beyond be.

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