Example 2

A discrete space which has more than one point is not connected.

Observe that if $(U_i)_{i \in I}$ is a partition of a topological space $X$ consisting of open sets [necessarily non-empty, from the definition of a partition (Set Theory, R, , no. 4)] then each of the $U_i$ is both open and closed in $X$, for the complement of $U_i$ is the union of the $U_x$ for which $x \neq i$. The open subsets of $X$ are the subsets $A$ such that $A \cap U_i$ is open in $U_i$ for each $i \in I$, so that $X$ can be identified with the sum of the $U_i$ ($\S 2$, no. 4, Example 3) and is not connected if $I$ has more than one element