Connectedness

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.

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 $c \in I$.)

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.

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.

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.

Proposition 6.4. The continuous image of a connected space is connected. Therefore connectedness is a topological invariant.

Proof. If $f : X \to Y$ is continuous and $X$ is connected, suppose $f(X) = U \cup V$ with $U, V$ open and disjoint in $f(X)$. Then $f^{-1}(U)$ and $f^{-1}(V)$ are open, disjoint, and cover $X$, contradicting connectedness. $\blacksquare$

Proposition 6.5 (Intermediate Value Theorem). If $f : X \to \mathbb{R}$ is continuous and $X$ is connected, then $f$ attains every value between any two of its values. This generalises the classical IVT from $\mathbb{R}$ to any connected space.

Proposition 6.6 (Products). The product of connected spaces is connected (with the product topology). Finite products follow from Proposition 6.4 by noting $X \times Y$ is homeomorphic to the image of the product space under a continuous map; arbitrary products require more care but also hold.

Proposition 6.7 (Closure). If $A \subseteq X$ is connected, then any set $B$ with $A \subseteq B \subseteq \overline{A}$ is connected. In particular, the closure of a connected set is connected.

Definition. A space is totally disconnected if its only connected subsets are singletons. Examples include $\mathbb{Q}$ (with the subspace topology from $\mathbb{R}$) and the Cantor set $C$.

Proposition 6.8. The Cantor set $C$ is totally disconnected, compact, uncountable, and perfect (every point is a limit point). It is homeomorphic to $\{0, 1\}^{\mathbb{N}}$ with the product topology.

Problem. Show that $\operatorname{GL}(2, \mathbb{R})$ (the set of $2 \times 2$ invertible real matrices with the subspace topology from $\mathbb{R}^4$) is disconnected.

Problem. For which $n$ is $\mathbb{R}^n \setminus \{0\}$ path-connected?