When Is a Continuous Bijection a Homeomorphism?

Abstract

Homeomorphisms are one of the main objects of study in topology and, hence, it is interesting to provide conditions that ensure that a map between topological spaces is a homeomorphism. Some examples of these kinds of results are the invariance of domain theorem (IDT) or the well-known lemma that states that a continuous bijection from a compact topological space to a Hausdorff topological space is a homeomorphism. In this note, a similar result is provided. More specifically, we show that any continuous bijection from a path-connected topological space to a set endowed with the order topology is a homeomorphism. In particular, we show how this generalizes the easiest case of the IDT for dimension n = 1. Furthermore, we apply the result in the context of special relativity and Lorentz transformations.