Homöomorphie

Homöomorphie is a concept in mathematics, specifically within topology, that describes a specific type of equivalence between topological spaces. Two topological spaces are considered homeomorphic if there exists a homeomorphism between them. A homeomorphism is a bijective function (a function that is both one-to-one and onto) between two topological spaces that is also continuous, and whose inverse function is also continuous.

In simpler terms, two topological spaces are homeomorphic if one can be continuously deformed into the other without tearing or gluing any parts. This means that they share fundamental topological properties, such as the number of holes, connectedness, and compactness. For example, a coffee cup and a donut are considered homeomorphic because one can be continuously transformed into the other. This is a classic example used to illustrate the concept, as both objects possess a single hole.

The existence of a homeomorphism is the defining characteristic of topological equivalence. If two spaces are homeomorphic, they are essentially the same from a topological perspective. This concept allows topologists to classify and understand different spaces by grouping together those that are topologically indistinguishable. This is a powerful tool for studying the qualitative properties of spaces that are invariant under continuous deformations.