Mathematical definition vs Geometric Explanation of Homeomorphism. How can we connect the mathematical definition of Homeomorphism with its geometric interpretation?
Lets say if I give a Cut what properties of Homeomorphism will be violated?

If a function introduces a cut, it is not continuous: continuous functions preserve connectedness. If a function glues, then its inverse is not continuous: functions with continuous inverses are open, meaning that they map open sets to open sets. Intuitively, if you glue two open sets together, there is a kind of boundary between the glued sets where they were glued together. Before gluing, these points weren't boundary points (since open sets dont contain their boundary points) while after gluing they are. For instance, $0$ is not a boundary point of $[0,1/2)$ as a subset of the topological space $[0,1)$, but $1=f(0)$ is a boundary point of $f([0,1/2))$, where $f(t)=\exp(2\pi\mathrm it)$ is a map to the unit circle in the complex plane which glues the ends of the line together. So this map is not open, since sets containing their own boundary points aren't open. And finally, if a function squishes points into each other, it is not injective, and thus not invertible.
