Homeomorphisms & Topological Equivalence

When two spaces count as the same, and the invariants that tell them apart.

The idea

A homeomorphism is topology's notion of sameness: two spaces count as the same when a reversible map between them respects the open sets, the structure a topological space carries.

Precisely, a homeomorphism $f \colon X \to Y$ is a bijection such that both $f$ and its inverse $f^{-1}$ are continuous. The bijection matches up the points; continuity of $f$ says open sets of $Y$ pull back to open sets of $X$; continuity of $f^{-1}$ says the same in the other direction. Spaces joined by such a map are homeomorphic, or topologically equivalent.

The condition on $f^{-1}$ is not automatic. Wrapping the half-open interval $[0, 2\pi)$ once around the circle is a continuous bijection, but its inverse jumps at the seam, so it is not a homeomorphism.

The definition mentions no distance, length, or angle, so a homeomorphism may stretch and bend freely; it may only never tear or glue. That is why the bounded interval $(-1, 1)$ is homeomorphic to the whole real line. A property that survives every homeomorphism is a topological invariant, and invariants prove non-equivalence: two spaces that disagree on one cannot be homeomorphic.

Ways to work on it

• Walkthrough. The definition, a concrete homeomorphism, and an invariant that blocks one.
• Practice. Decide whether two spaces are homeomorphic.
• Hardest. Separate two confusable spaces with a finer invariant.

Not sure where to start? Take the ten-question placement test.