Homeomorphism

Homeomorphism

A blue trefoil knot, an example of a topological space.

In the field of topology, a homeomorphism is a continuous function between two topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces; they are the mappings that preserve all the topological properties of a space.

A homeomorphism can be thought of as a "stretching" or "bending" of a space into another without tearing or gluing. Two spaces that are homeomorphic are considered to be topologically equivalent.

Definition

Let \(X\) and \(Y\) be topological spaces. A function \(f: X \to Y\) is a homeomorphism if it satisfies the following conditions:

1. Bijective: \(f\) is a bijective function, meaning it is both injective and surjective.
2. Continuous: \(f\) is a continuous function.
3. Inverse is continuous: The inverse function \(f^{-1}: Y \to X\) is also continuous.

If such a function exists, \(X\) and \(Y\) are said to be homeomorphic, and we write \(X \cong Y\).

Examples

• The circle \(S^1\) is homeomorphic to any ellipse in the plane. This is because an ellipse can be continuously deformed into a circle without cutting or gluing.
• The surface of a sphere is homeomorphic to the surface of any ellipsoid.
• The Möbius strip is not homeomorphic to a cylinder, as they have different topological properties.

Properties

Homeomorphisms preserve topological properties such as:

• Connectedness: If a space is connected, any space homeomorphic to it is also connected.
• Compactness: A compact space remains compact under a homeomorphism.
• Hausdorff property: If a space is Hausdorff, any space homeomorphic to it is also Hausdorff.

Applications

Homeomorphisms are fundamental in topology because they allow mathematicians to classify spaces based on their topological properties rather than their geometric shape. This concept is crucial in areas such as:

• Algebraic topology, where spaces are studied up to homeomorphism.
• Differential topology, where smooth structures are considered up to diffeomorphism, a type of homeomorphism.
• Knot theory, where knots are studied as embeddings of circles in 3-dimensional space, up to ambient isotopy, a form of homeomorphism.

Related pages

• Topological space
• Continuous function
• Isomorphism
• Diffeomorphism
• Knot theory