Connectedness

Let $X$ be a topological space. A separation of $X$ is a pair of disjoint non-empty open sets whose union is $X.$

Let $X$ be a topological space. The space $X$ is disconnected if and only if it has a separation. The space $X$ is connected if and only if it is not disconnected. Equivalently, $X$ is connected if and only if the only subsets of $X$ that are both open and closed are $\varnothing$ and $X.$

Let $X$ be a topological space and $A\subseteq X.$ The set $A$ is connected if and only if it is connected as a subspace of $X.$ Connected sets have the following properties:

• a subset of $\mathbb{R}$ is connected if and only if it is an interval;
• the rational numbers are disconnected, since $\left\{x\in\mathbb{Q}:x<\sqrt{2}\right\}$ and $\left\{x\in\mathbb{Q}:x>\sqrt{2}\right\}$ separate them;
• a set with the discrete topology is connected only if it has at most one point;
• the closure of a connected set is connected, as is any set between a connected set and its closure;
• the union of connected sets with a common point is connected;
• the product of finitely many connected spaces is connected;
• the continuous image of a connected set is connected: if $f:X\rightarrow Y$ is continuous and $A\subseteq X$ is connected then $f(A)$ is connected.

Since the connected subsets of $\mathbb{R}$ are the intervals, a continuous real-valued function on a connected space takes every value between any two of its values, which extends the intermediate value theorem. Connectedness is a topological property, so an interval is not homeomorphic to a circle, which stays connected when a point is removed.

Let $X$ be a topological space and $x\in X.$ The connected component of $x$ is the union of all connected subsets of $X$ containing $x.$ The connected component of $x$ is the largest connected set containing $x.$ Connected components have the following properties:

• the connected components partition $X$ into closed sets;
• $X$ is connected if and only if it has exactly one connected component;
• the connected components of $\mathbb{Q}$ are its single points.

Let $X$ be a topological space and $x,y\in X.$ A path in $X$ from $x$ to $y$ is a continuous function $\gamma:\left[0,1\right]\rightarrow X$ with $\gamma(0)=x$ and $\gamma(1)=y.$

Let $X$ be a topological space. The space $X$ is path-connected if and only if any two of its points are joined by a path.

Let $X$ be a topological space and $x\in X.$ The path component of $x$ is the set of points of $X$ that are joined to $x$ by a path.

Let $\underline{x},\underline{y}\in\mathbb{R}^n.$ The line segment between $\underline{x}$ and $\underline{y}$ is the set $\left\{\left(1-t\right)\underline{x}+t\,\underline{y}:t\in\left[0,1\right]\right\}.$ Path-connected sets have the following properties:

• a subset of $\mathbb{R}^n$ that contains the line segment between any two of its points is path-connected by those segments, so balls, intervals and $\mathbb{R}^n$ are path-connected;
• spheres and $\mathbb{R}^n$ with a point removed are path-connected when $n\geq 2;$
• the continuous image of a path-connected set is path-connected;
• a path-connected space is connected, since the image of each path is connected and the paths from a fixed point share that point.

The converse of the last property is false. The topologist’s sine curve is the closure in $\mathbb{R}^2$ of the graph of $x\mapsto\sin\frac{1}{x}$ on $\left(0,1\right].$ The topologist’s sine curve is connected, but no path joins the graph to the segment $\left\{0\right\}\times\left[-1,1\right]$ in its closure. An open connected subset of $\mathbb{R}^n$ is path-connected, and any two of its points are joined by a path made of finitely many line segments, so connectedness and path-connectedness agree for the open sets on which vector fields are defined.