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
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.
What you said is stronger than being homotopic. homotopic is weaker, for instance a line is homotopic to a point, By taking the straw (even if it has thickness) and just shrinking it along its longer axis you eventually arrive at a circle. If it has thickness you will arrive at a band and then you can also retract radially to arrive at a circle.
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
I think people don’t know a torus is hollow.
You are talking about a straw of zero wall thickness right? A real straw should be homo-whatever to a torus
Homotopic: Having the same (homo-) topological properties (-topic)
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.
Wouldn’t a straw be the product of a circle and a line?
What you said is stronger than being homotopic. homotopic is weaker, for instance a line is homotopic to a point, By taking the straw (even if it has thickness) and just shrinking it along its longer axis you eventually arrive at a circle. If it has thickness you will arrive at a band and then you can also retract radially to arrive at a circle.
A CD is clearly homotopic to a torus, though…
And the walls of a straw do have thickness…
A straw goes:
Gas - solid - gas - solid - gas
If solid torus yes, if just the regular torus (surface of the solid torus) no. CD is homotopic to a circle and so is a solid torus.
OK, that’s my ignorance. I didn’t realise toruses were usually hollow.
Thank you for letting me know, you’re right and I’ve learnt something.