Definition:Homeomorphism

This page is about homeomorphisms, the isomorphisms in topology. For other uses, see Definition:Isomorphism.

Definition

Topological Spaces

Let $T$ and $T'$ be topological spaces.

Let $f: T \to T'$ be a bijection such that both $f$ and $f^{-1}$ are continuous.

Then $f$ is a homeomorphism. We can say that $T$ and $T'$ are homeomorphic.

The symbol $T \sim T'$ is often seen.

Metric Spaces

Let $M$ and $M'$ be metric spaces.

Let $f: M \to M'$ be a bijection such that both $f$ and $f^{-1}$ are continuous.

Then $f$ is a homeomorphism.

This definition also follows directly from:

• The fact that a metric space induces a topology;
• Equivalence of Metric Space Continuity Definitions.

Manifolds

A homeomorphism of a manifold $X$ to a manifold $Y$ is a continuous bijection such that the inverse is also continuous.

Equivalent Definitions

• By definition of continuity, a homeomorphism is a bijection $f: T \to T'$ such that $U$ is open in $T$ iff $f \left({U}\right)$ is open in $T'$.

• By Bijection is Open iff Inverse is Continuous a homeomorphism is a bijection which is both open and continuous.

• By Bijection is Open iff Closed it follows that a homeomorphism is a bijection which is both closed and continuous.

Also known as

Also known as:

• a topological equivalence, usually used when the spaces in question are metric spaces
• an isomorphism.

Caution

Not to be confused with homomorphism.

Also see

• Inverse of Homeomorphism is Homeomorphism

• Results about Homeomorphisms can be found here.

Sources

• Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): $\S 1.1$: Definition $2$
• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Functions
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Definition $2.4.7$