Connected and Disconnected Sets¶

Definition¶

Consider a space $X$ in which openness is defined. A subset $E \subseteq X$ is called disconnected if it has an open cover $\{V,W\}$ such that $E \subseteq V\cup W$ and the intersections $E\cap V$ and $E\cap W$ are disjoint and nonempty. Thus $E\cap V\cap W=\varnothing$ $\emptyset$ ; the ambient sets $V$ and $W$ may overlap outside $E$ . If $E$ is nonempty and not disconnected, it is connected.

Interpretation¶

A set is disconnected if it can be split into two nonempty pieces, each cut out by an open set, with no point of the set in both. A set is connected if it comes in one piece, so that no such split exists.

Two panels: a set in two pieces covered by two disjoint open sets V and W, so it is disconnected, and a single blob that cannot be split this way, so it is connected.

On the left, the dark set $E$ lies in the union of the open sets $V$ and $W$ . It meets both, and no point of $E$ lies in both, so it is disconnected. The right-hand set cannot be split in this way: whenever two open sets cover it and each meets it, they share a point of the set, so it is connected.

A common slip is to read "split" as "the pieces do not touch". The set $(0,1)\cup(1,2)$ is disconnected, with $V=(0,1)$ and $W=(1,2)$ $(1, 2)$ , although its two pieces come arbitrarily close at $1$ . On the other hand, $(0,1]$ and $(1,2)$ are disjoint pieces of the connected set $(0,2)$ : every open set containing $1$ also contains points of $(1,2)$ , so no open set cuts out $(0,1]$ .

In $\mathbb{R}$ the connected sets are exactly the intervals, which is why the intermediate value theorem of Math I is stated on intervals; Math III extends it to the intermediate value theorem on connected sets. Connectedness is often proved through path-connectedness, which implies it and is easier to check.

Key Results¶

The only type of connected subsets of $\mathbb{R}$ is interval.
Consider metric spaces $(X,d_{X})$ and $(Y,d_{Y})\text{.}$ Let $f\colon X\to Y$ be a continuous function, and let $E$ be a connected subset of $X\text{.}$ Then $f(E)$ is a connected subset of $Y\text{.}$ This result is known as continuity preserves connectedness.

Examples¶

Any singleton set is connected.
$\mathbb{Q}$ is a disconnected subset of $\mathbb{R}$ .
In a discrete metric space, any set with at least two points is disconnected because each singleton is open.

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 10

Aliases: connected set, disconnected set Keywords: connectedness, disconnectedness Symbols: intersection, rational numbers, real numbers, subset, union

Last updated: 20 October 2026 03:02