we know that the following shapes are homeomorphic
but in my book write :
Geometrically speaking, a homeomorphism is a bijection that can bend, twist, stretch, and wrinkle the space M to make it coincide with N, but it can not rip, puncture, shred, or pulverize M in the process.
now my question is :
we know that the above shapes are homeomorphic , but we $ \ can't $ bend, twist, stretch, and wrinkle the space M to make it coincide with N. what is wrong ? what is relation between this question and homotopy and knot theory ?
A better intuition for homeomorphic is that you are allowed to cut it apart, stretch it any any way, as long as you glue it back together with the same parts touching afterwards.
In the case of knots, they are all homeomorphic since you can always cut the string, unknot it, and then glue it back together to make a circle.
Knots are usually classified up to an "ambient isotopy", which means you are allowed to stretch the knot, but not cut it up at all, or let it pass through itself.
This is all pre-rigorous thinking though. You really need to get practice carefully applying the definitions first.