Connected Set — Definition, Formula & Examples

A connected set is a set that cannot be divided into two nonempty, disjoint open subsets. Informally, it is "all in one piece" — there is no way to separate it into two parts without breaking continuity.

A topological space $X$ is connected if there do not exist two nonempty open sets $U$ and $V$ such that $U \cap V = \emptyset$ and $U \cup V = X$ . A subset $S$ of a topological space is connected if it is connected as a subspace with the subspace topology.

How It Works

To determine whether a set is connected, you look for a way to partition it into two nonempty open sets that cover the entire space. If no such partition exists, the set is connected. In

\mathbb{R}

with the standard topology, the connected subsets are exactly the intervals (including rays and

\mathbb{R}

itself). A set like

(0,1) \cup (2,3)

is disconnected because the two intervals form a separation. Continuous functions preserve connectedness: if

f

is continuous and

S

is connected, then

f(S)

is also connected — this fact underlies the Intermediate Value Theorem.

Example

Problem: Determine whether the subset

S = (0, 1) \cup (2, 3)

\mathbb{R}

is connected.

Step 1: Try to find a separation: two nonempty, disjoint open sets in the subspace topology whose union is

S

U = (0,1), \quad V = (2,3)

Step 2: Check the conditions. Both

U

and

V

are open in the subspace topology on

S

, both are nonempty, they are disjoint, and their union equals

S

U \cap V = \emptyset, \quad U \cup V = S

Step 3: Since a valid separation exists, the set is not connected.

Answer:

S = (0,1) \cup (2,3)

is disconnected.

Why It Matters

Connectedness is the topological property behind the Intermediate Value Theorem: a continuous function on a connected domain cannot "skip" values. It appears throughout real analysis, complex analysis (where connected open sets are called domains), and algebraic topology when classifying spaces by their connected components.

Common Mistakes

Mistake: Confusing connected with path-connected. Students sometimes assume that if a set is connected, you can always draw a continuous path between any two points.

Correction: Path-connected implies connected, but the converse is not always true. The topologist's sine curve is a classic example of a connected set that is not path-connected.