# Definition Given two [[Topological Space|topological spaces]] $X_1$ and $X_2$, a homeomorphism $f:X_1 \rightarrow X_2$ is a [[Types of Maps|bijection]] from one space onto another which is [[continuous map|continuous]] and whose [[Inverse Map|inverse]] is also continuous. Intuitively speaking, we say two spaces are homeomorphic if we can *continuously* deform them into one another, i.e., without cutting and gluing, poking holes or closing holes. Note that homeomorphisms form an [[Equivalence Relation]], thus it they divide all topological spaces into [[Equivalence Class|equivalence classes]] according to whether they can be continuously deformed into one another. We have no general way of characterizing the equivalence classes of homeomorphisms, but we can use [[Topological Invariant|topological invariants]] to say that if two topological spaces have different topological invariants, then they cannot be homeomorphic to one another. # Examples ![[Pasted image 20210630183916.png]] These objects are homeomorphic. The connected rings can be disentangled in $\mathbb{R}^4$ so they are homeomorphic to the disconnected rings. The [[torus]] $T^2$ and the coffee mug are in the same homeomorphism equivalence class.