Homeomorphism

In mathematics, a homeomorphism is a function that maps one topological space to another with the property that it is bijective and both the function and its inverse are continuous with respect to the associated topologies. A homeomorphism indicates that the two topological spaces are "geometrically" alike, in the sense that points that are "close" in one space are mapped to points which are also "close" in the other, while points that are "distant" are also mapped to points which are also "distant". In differential geometry, this means that one topological space can be deformed into the other by "bending" and "stretching".

Formal definition

Let $\scriptstyle (X,O_{X})$ and $\scriptstyle (Y,O_{Y})$ be topological spaces. A function $\scriptstyle f:(X,O_{X})\rightarrow (Y,O_{Y})$ is a homeomorphism between $\scriptstyle (X,O_{X})$ and $\scriptstyle (Y,O_{Y})$ if it has the following properties:

1. f is a bijective function (i.e., it is one-to-one and onto)
2. f is continuous
3. The inverse function $\scriptstyle f^{-1}:(Y,O_{Y})\rightarrow (X,O_{X})$ is a continuous function.

If some homeomorphism exists between two topological spaces $\scriptstyle (X,O_{X})$ and $\scriptstyle (Y,O_{Y})$ then they are said to be homeomorphic to one another. Homeomorphism (in the sense of being homeomorphic) is an equivalence relation.

Topological property

A topological property is one which is preserved by homeomorphism. Examples include

• Compactness;
• Connectedness.