Homeomorphism

Introduction

In the field of topology, a branch of mathematics, homeomorphism is a fundamental concept that defines the equivalence of topological spaces. A homeomorphism, also known as a bicontinuous function, is a bijective and continuous function between two topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space.

Definition

Formally, a function f : X → Y between two topological spaces (X, T) and (Y, S) is called a homeomorphism if it has the following properties:

1. f is a bijection (one-to-one and onto), 2. f is continuous, and 3. The inverse function f^-1 is continuous.

If such a function exists, we say X and Y are homeomorphic, and X is a homeomorph of Y. A self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form an equivalence relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism classes.

A photograph of two different shapes that are homeomorphic, such as a coffee mug and a donut.

Properties

Homeomorphisms are respected by other topological properties. For instance, if X and Y are homeomorphic, then:

1. X is compact if and only if Y is compact. 2. X is connected if and only if Y is connected. 3. X is Hausdorff if and only if Y is Hausdorff. 4. X is path connected if and only if Y is path connected.

This is because homeomorphism is the most refined notion of equivalence in topology: it respects all possible topological properties.

Examples

The classic example of a homeomorphism is the "coffee cup and the donut" scenario. In this case, a coffee cup with a handle and a donut are homeomorphic to each other. This is because one can be deformed into the other without tearing or gluing. This property—being able to deform an object continuously into another—characterizes the essence of topological spaces.

Another example is the open interval (0, 1), which is homeomorphic to the real numbers R. The function f : (0, 1) → R defined by f(x) = (2x - 1) / (x - x²) is a homeomorphism between these two spaces.

Homeomorphisms in Analysis

In real analysis, homeomorphisms are integral to the study of conformal maps, and provide a formal way to study the similarities and differences between different mathematical spaces. In complex analysis, a branch of mathematical analysis, conformal maps are defined as functions that are locally homeomorphic to the open unit disk.

Homeomorphisms in Algebraic Topology

In algebraic topology, another branch of mathematics, homeomorphisms are used to define the fundamental group of a topological space. The set of all self-homeomorphisms of a space forms a group under the operation of function composition, and this group can be used to derive many interesting properties of the space.