Definition of connectedness? Connectedness is defined as: "A metric space $E$ is connected if the only subsets of $E$ which are both open and closed are $E$ and $\varnothing$. A subset $S$ of a metric space is a connected subset if the subspace $S$ is connected."
Can someone provide me with a more trivial/simple definition of connectedness?

Topological space is connected if and only if the only subsets in it, simultaneously open and closed, are the space itself and the empty subset.
In case of a metric space, the topology is induced by the metric.
As a counter-example, consider say a pair of parallel lines or planes as a single topological space - it is not connected.
Note also that it is not the same as path-connectedness.
And that the quality of being (dis)connected is not an intrinsic property of s (sub)set but is relative to the topological space with relation to which it is considered.
