Connectedness

Definition

A topological space $X$ is connected if it cannot be written as the disjoint union of two nonempty open sets.

Equivalently, the only clopen (both closed and open) subsets are $\varnothing$ and $X$.

Path Connectedness

A space $X$ is path connected if for any $x,y\in X$, there exists a continuous map $\gamma :[0,1]\to X$ with $\gamma (0)=x$ and $\gamma (1)=y$.

$\text{Path connected}⟹\text{Connected}$

The converse is false in general (e.g., the topologist’s sine curve).

Components

Every space $X$ decomposes uniquely into connected components — maximal connected subspaces.