quasar987 said:The usual tool for proving the "no homeo thm" is Brouwer's Invariance of Domain theorem:
http://en.wikipedia.org/wiki/Invariance_of_domain
It can in the same way be used to answer your question: Assume a homeo btw S (subset of R^n) and R^m exists. Consider R^n as a subset of R^m (say as R^n x {0,...,0}). Then we have a map
R^m --> S --> R^m
which is the homeomorphism of R^m with S composed with the inclusion of R^n in R^m. This map is not open since the inclusion of R^n in R^m maps any subset of R^n to a non open subset of R^m. This contradicts Brouwer's invariance of domain theorem.
A homeomorphism is a mathematical concept that describes a continuous mapping between two topological spaces that preserves their structure. In simpler terms, it is a function that can transform one space into another without tearing or gluing any points together.
A homeomorphism and an isomorphism are both types of mathematical mappings, but they differ in the types of structures they preserve. A homeomorphism preserves the topological structure of a space, while an isomorphism preserves algebraic or geometric structures.
Homeomorphisms play a crucial role in topology, which is the study of the properties of geometric shapes that are preserved under continuous deformations. They provide a way to compare and classify different spaces, and are also used in many other areas of mathematics, including differential equations and complex analysis.
One example of a homeomorphism is the mapping between a circle and a square. By continuously stretching and bending the circle, it can be transformed into a square without tearing or gluing any points. This demonstrates how the topological structure of both shapes is preserved by the homeomorphism.
No, not all homeomorphisms are reversible. A homeomorphism is only reversible if it has an inverse function that also satisfies the definition of a homeomorphism. In other words, if the inverse function can transform the square back into a circle without tearing or gluing any points, then the homeomorphism is reversible.