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Connectedness

Definitions

A topological space $(X, \mathbb{T})$ is disconnected if there are disjoint nonempty open sets $U,V \subset X$ such that $X = U \cup V$. A space is connected if it is not disconnected.

A subset $A \subset X$ is connected if it is connected as a subspace, or alternatively if there do not exist open and disjoint nonempty sets $U,V \subset X$ whose union contains $A$.

Properties

The image of a connected set/space under a continuous map is a connected set/space.

all these pages adapted with probably insufficient credit from my university's lecture notes