Proposition 4

Let $A$ be a connected subset of a topological space $X$, and let $f$ be a continuous mapping of $X$ into a topological space $X'$. Then $f(A)$ is connected.

Suppose $f(A)$ is not connected. Then there exist two sets $M', N'$ which are open in $f^{-1}(A)$ and which form a partition of $f(A)$; hence $A \cap f^{-1}(M')$ and $A \cap f^{-1}(N')$ are open in $A$ and form a partition of $A$; this contradicts the hypothesis that $A$ is connected.

The inverse image of a connected set under a continuous mapping need not be connected; consider for example a mapping of a discrete space into a space consisting of one point.

From Proposition 4 we derive another characterization of non-connected spaces: