connected space

A metric space is said to be disconnected if it is the union of two disjoint non-empty open set. Otherwise, X is said to be connected.

In other word, if we can find two open sets and such that and , then is disconnected.

Subtopics

• path connectivity

Connectness on a Real line

The following statements are equivalent for

• is connected
• s.t. , is contained in X
• is an interval

Continuity Preserves Connectedness

Let be a continuous map and be a connected subset of , then the image is continuous.

We can use this property to prove the intermediate value theorem.