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  • iAvicenna@lemmy.worldtoScience Memes@mander.xyzBaldur's Gayte
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    2 days ago

    Even if it has thickness still homotopic to a circle. For instance a band with thickness is homotopic to a circle, you can retract along the radius to arrive at a circle that is inside the band. Similarly a plane, or a slab with thickness are all homotopic to a point.

    Note that all of these are proved by using collections of transformations from the space to itself (not necessarily from the space to all of itself though, if it maps the space to a subset of it that is fine). So if you want to say something like “but you can also shrink a circle to eventually reach a point but it is not homotopic to a point” that won’t work because you are imagining transformation that maps a circle not into itself but to a smaller one.

    ps: the actual definition of homotopy equivalence between “objects” is slightly more involved but intuitively it boils down to this when you imagine one space as a subset of the other and try to see if they are homotopy equivalent.





  • iAvicenna@lemmy.worldtoScience Memes@mander.xyzBaldur's Gayte
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    2 days ago

    a torus is not homotopic to a straw though unless you take the straw and glue it at its ends. a straw is homotopic to a circle, a torus is homotopic to product of two circles, Baldur’s gate is homotopic to a disk which is homotopic to a point unless we are talking about the game storage medium which used to be a CD which is also homotopic to a circle