In topology,we are often told that homeomorphism can be thought of as a continuous deformation of a shape. I mean by continuous deformation the actions of twisting, bending, stretching but not tearing or gluing.
My question is how can I show formally that these actions satisfy the definitions of homeomorphism. Suppose we are asked to check a continuous surjection from a shape to another. Does it suffice if I could achieve the second from the first by twisting, stretching and bending and not tearing. Even we are allowed to glue (identify two points) here, because we are required to check continuous surjection, so injectivity may be violated. But I want to make it clear to me that these vague actions actually are continuous as maps. How can I show that.