I've read that
geometrically speaking, a homeomorphism from $M$ to $N$ is a bijection that can bend, twist, stretch and wrinkle the space $M$ to make it coincide with $N$ but it cannot rip, puncture, shred or pulverize $M$ in the process.
I have difficulty to see why a continuous bijection (with continuous inverse) can be seen as doing such things. Can someone elaborate or give me ways to visualize it?