Please clear my confusion about difference between homotopy, topology and isotopy. The first question is: Is it true two objects are isotopic implies they are topologically equivalent and topologically equivalent implies they are homotopic? Next question: Let C be a circle. Let C1 be C subtracted a set of a point p and C2 be C subtracted a set of a line segment L. Is C1 isotopic,topologically equivalent or homotopic to C2 respectively?
Thank you in advance.
A.O
If by topologically equivalent you mean homeomorphic then this is stronger than being isotopy equivalent or homotopy equivalent. For example, $\mathbb R^n$ and $\mathbb R^m$ are homotopy equivalent (they are both contractible : homotopy equivalent to a point). But $\mathbb R^n$ is not homeomorphic to $\mathbb R^m$ if $m\not =n$. Being isotopy equivalent is stronger than homotopy equivalent as we put some further restrictions on the isotopy equivalences.