< [[nlab:compact connected space]] ## Definition ## A **compact connected space** is a space $S$ such that for [[subtype|subspace]]s $A \subseteq S$ and $B \subseteq S$ of $S$ with canonical [[monic]] mappings $i_{A,S}:A \to S$ and $i_{B,S}:B \to S$ such that the canonical monic mapping $i_{A \cup B,S}:A \cup B \to S$ is an [[equivalence]] and the canonical monic mapping $i_{\emptyset,A \cap B}:\emptyset \to A \cap B$ is an [[equivalence]], either $i_{A,S}$ is an equivalence or $i_{B,S}$ is an equivalence. ## Examples ## * Assuming [[axiom R-flat]], the [[Dedekind real numbers]] are a compact connected space. ## See also ## * [[axiom R-flat]] * [[Dedekind real numbers]] ## References ## * [[Mike Shulman]], Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))