Example

In any topological space, the empty set and every set consisting of a single point are connected. In a Hausdorff space $X$, every finite set consisting of more than one point is not connected, and more generally every subset of $X$ which has more than one point and which has at least one isolated point is not connected.

If a dense subset $A$ is connected, then the whole space $X$ is connected; otherwise there would exist two non-empty disjoint open subsets $M, N$ of $X$ such that $M \cup N = X$, and $M \cap A, N \cap A$ would be two disjoint non-empty open subsets of $A$ whose union is $A$. Hence we have