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.

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.

Path Connectivity

A metric space is path connected Let be a subset of the metric space . We say that is path connected if for all , there is a function such that and .

A path connected set must be connected, but the converse is not true, counter examples include “Topologist’s sine curve”.