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”.