>[!quote] In a Nutshell >Grasps the abstract notion of how together a [[Space|space]] is, i.e. if we can move from one point to another without leaving the space. --- >[!info] Connected Topological Space >A topology is connected (**zusammenhängend**), if it cannot be represented as a union of multiple disjoint non-empty open subsets (A). If it can be represented like that (B), it is disconnected. ![[Pasted image 20240718174434.png|center|300]] >[!info] Connected Component >The connected component (**Zusammenhangskomponente**) of a point $x \in X$ is the union of all subsets that contain $x$ and therefore the unique largest connected subset of $X$ that contains $x$ (by definition of topology). TODO - sketch >[!info] Path Connectedness >A stronger notion of connectedness that requires the structure of a path from point $x$ to point $y$ in $X$, which is a [[Maps and Induced Structures - Functions, Pushforwards and Pullbacks|continuous function]] from the unit intervall to $X$. It induces a [[Equivalence Relation and Class|equivalence relation]], which makes the points equivalent if such a path exists. TODO - sketch >[!info] Simply-Connected Space >A simply- or 1-connected (**einfach zusammenhängend**) topological space is a path connected topological space, where every path can be continuously transformed into a point, it is homotope to a point.