Q1: Are the Fig (a) and (b), the equivalence "=" is Homeomorphic or Homotopic?
ps. for details of the figures see Ref here.
I learned that "The characterization of a homeomorphism often leads to confusion with the concept of homotopy, which is actually defined as a continuous deformation, but from one function to another, rather than one space to another. In the case of a homeomorphism, envisioning a continuous deformation is a mental tool for keeping track of which points on space X correspond to which points on Y—one just follows them as X deforms. In the case of homotopy, the continuous deformation from one map to the other is of the essence, and it is also less restrictive, since none of the maps involved need to be one-to-one or onto. Homotopy does lead to a relation on spaces: homotopy equivalence."
Q2: Does Homeomorphic necessary imply the Homotopic equivalence of space? So Homeomorphic is stronger restricted condition than Homotopic?