Connectedness

6.1 Connected and Disconnected Spaces

Definition. A topological space $X$ is disconnected if there exist nonempty disjoint open sets $U, V$ with $X = U \cup V$. Such a pair $\{U, V\}$ is called a separation of $X$.

$X$ is connected if it is not disconnected.

Equivalently, $X$ is connected if and only if the only clopen subsets of $X$ are $\emptyset$ and $X$.

Example 6.1. $[0, 1]$ is connected. $[0, 1) \cup (2, 3]$ is disconnected (with the subspace topology from $\mathbb{R}$).

Example 6.2. $\mathbb{Q}$ with the subspace topology from $\mathbb{R}$ is totally disconnected: the only connected subsets are singletons.

6.2 Connected Subsets of $\mathbb{R}$

Theorem 6.1. A subset of $\mathbb{R}$ (with the standard topology) is connected if and only if it is an interval.

(Here an interval is any set $I \subseteq \mathbb{R}$ with the property: if $a, b \in I$ and $a < c < b$, then $c \in I$.)

6.3 Path-Connectedness

Definition. A space $X$ is path-connected if for any two points $x, y \in X$, there exists a continuous function $\gamma : [0, 1] \to X$ with $\gamma(0) = x$ and $\gamma(1) = y$.

Proposition 6.1. Every path-connected space is connected. The converse is false.

Example 6.3 (Topologist”s sine curve). Let

$S = \{(x, \sin(1/x)) : 0 < x \leq 1\} \cup \{(0, y) : -1 \leq y \leq 1\} \subseteq \mathbb{R}^2.$

$S$ is connected but not path-connected.

Example 6.4. $\mathbb{R}^n$ is path-connected for all $n \geq 1$. Any convex subset of $\mathbb{R}^n$ is path-connected.

6.4 Components

Definition. A connected component of $X$ is a maximal connected subset of $X$. The connected components of $X$ form a partition of $X$.

Proposition 6.2. Connected components are closed (in a Hausdorff space, they are always closed).

Definition. A path component of $X$ is a maximal path-connected subset. Path components also partition $X$, and each path component is contained in a connected component.

6.5 Local Connectedness

Definition. $X$ is locally connected if for every $x \in X$ and every open neighbourhood $U$ of $x$, there exists a connected open neighbourhood $V$ of $x$ with $V \subseteq U$.

Proposition 6.3. Every open subset of $\mathbb{R}^n$ is locally connected.

Example 6.5. The topologist’s sine curve is connected but not locally connected.