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.

中文翻译：

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什么时候连续双射是同胚？

摘要 同胚是拓扑学的主要研究对象之一，因此，提供确保拓扑空间之间的映射是同胚的条件是很有趣的。这些结果的一些例子是域定理的不变性 (IDT) 或众所周知的引理，该引理表明从紧拓扑空间到 Hausdorff 拓扑空间的连续双射是同胚。在本说明中，提供了类似的结果。更具体地说，我们证明了从路径连接的拓扑空间到具有序拓扑的集合的任何连续双射都是同胚。特别是，我们展示了这如何推广维度 n = 1 的 IDT 的最简单情况。此外，我们将结果应用于狭义相对论和洛伦兹变换的上下文中。