Connectedness

Separations, clopen sets, and the topological route to the Intermediate Value Theorem.

The idea

Connectedness makes precise the idea that a space is all one piece. Every set can be split into two parts; what matters is whether the two parts touch, and open sets are what make touching precise.

Definition (Connected space).

A separation of a space $X$ is a pair of disjoint nonempty open sets $U$ and $V$ with $X = U \cup V$. The space $X$ is connected when no separation of it exists.

Openness gives every point of each piece room to spare inside its own piece, so no point of $U$ is approached by $V$ and no point of $V$ by $U$: the two halves genuinely fall apart. In a separation each piece is the complement of the other, so each is open and closed at once — clopen. Every space has two clopen subsets, $\varnothing$ and itself; connectedness says it has no others.

The definition is negative — connected means a certain object does not exist — which makes it hard to verify directly but powerful to apply. Its main consequence is the following theorem.

Theorem (Continuous images of connected spaces).

If $f \colon X \to Y$ is continuous and $X$ is connected, then the image $f(X)$ is connected.

Since the connected subsets of $\mathbb{R}$ are exactly the intervals, this theorem yields the Intermediate Value Theorem.

Ways to work on it

• Walkthrough. Separations, the clopen characterization, and continuous images.
• Practice. Decide whether a subspace of the real line is connected.
• Hardest. Use connectedness to force a root via the Intermediate Value Theorem.

Not sure where to start? Take the ten-question placement test.