Homeomorphism

A bijective continuous map between topological spaces whose inverse is continuous. It identifies spaces that have the same topological structure, regardless of their geometric shape.

Topological space: A set equipped with a collection of subsets, called open sets, satisfying axioms that define its topology. Homeomorphisms compare spaces through their open sets rather than distances.
Open map: A function that sends every open set in its domain to an open set in its codomain. A bijective continuous map is a homeomorphism exactly when it is also open.
Circle: The set of points in a plane at a fixed distance from a center. A circle and the boundary of a square are homeomorphic despite their different outlines.
Homotopy equivalence: A pair of continuous maps whose composites are continuously deformable to identity maps. It is weaker than homeomorphism: spaces can be homotopy equivalent without being homeomorphic.
Topological invariant: A quantity or property unchanged under homeomorphism. Invariants distinguish spaces that cannot be homeomorphic.
Continuous function: A function between topological spaces whose preimage of every open set is open. Continuity is one of the two conditions a homeomorphism must satisfy.
Closed map: A function that sends every closed set in its domain to a closed set in its codomain. For a bijection, closedness can provide an alternative test for continuity of the inverse.
Square: A quadrilateral with four equal sides and four right angles. Its boundary has the same topological structure as a circle.
Diffeomorphism: A smooth bijection between smooth manifolds whose inverse is smooth. It requires differentiable structure and is stronger than topological equivalence.
Connected space: A topological space that cannot be separated into two disjoint nonempty open subsets. Connectedness is preserved by homeomorphisms.

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