Invariance of domain

Imagine you have a squishy ball that you can stretch and bend in any way you like, but you can't tear it or make any holes. Now imagine you have two different balls: one that's very skinny and long like a hotdog, and another that's short and round like a basketball.

Invariance of domain is a rule that says you can never change the shape of either of these balls so much that the skinny hotdog ball becomes the short round basketball ball, or vice versa. No matter how much you stretch or squish or bend the balls, their shapes will always stay the same - skinny and long or short and round.

This rule applies to shapes that are more complicated than just balls too, like shapes in really high dimensions that you can't even see or imagine. But no matter how complicated a shape is, if it has the property of invariance of domain, then it will always keep its shape no matter how much you try to change it.