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Let $X, Y$ be Topological Spaces and let $f: X \to Y$ be a Function. $f$ is called a Homeomorphism if

1. $f$ is a Bijection
2. $f$ is a Continuous Function
3. $f^{-1}$ is a Continuous Function.

If such an $f$ exists between two spaces, we say they are homeomorphic and write $X \sim Y$.

Spaces that are homeomorphic share many properties such as

1. Compactness is a Topological Invariant
2. Connectedness is a Topological Invariant

It is not hard to see that $\sim$ is an Equivalence Relation between Topological Spaces. Proof TODO.