What is the mathematical word to describe when two objects can be transformed to each other like Klein bottle and torus?

Question

In mathematics how does one say that two objects like the Klein bottle and a torus can be transformed into each other and are the same thing in some sense?

Answer 1

Expanding my comment into an answer: The term is homeomorphic. In particular, we call two topological spaces homeomorphic if there exists $f:X\to Y$ such that $f$ is a continuous bijection and $f^{-1}$ is continuous.

Answer 2

There are a few notions depending on what type of equivalence you want to consider.

• A diffeomorphism of two manifolds $M$ and $N$ is a smooth bijection $f:M\to N$ with smooth inverse.

• Homeomorphism of two topological spaces $X$ and $Y$ is when there exists a map $f:X\to Y$ which is a continuous bijection with continuous inverse.

• A homotopy equivalence of spaces is a pair of maps $f:X\to Y$ and $g:Y\to X$ with $g\circ f\simeq \operatorname{Id}_X$ and $f\circ g \simeq \operatorname{Id}_Y$.

Diffeomorphism $\implies$ Homeomorphism $\implies$ Homotopy Equivalence.

There are also other notions of equivalence like isometry, etc.