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?
 A: A topological space $X$ is connected if for open sets $U,V\subseteq X$ it is not possible to have all of the following conditions:


*

*$X=U\cup V$

*$U\cap V=\emptyset$

*$U\ne \emptyset$

*$V\ne \emptyset$


A subset $S$ of a topological space $X$ is connected if it is connected as a topological space with the subspace topology.
A: A subset $S$ of a metric space is disconnected if it can be partitioned into two disjoint nonempty subsets $A$ and $B$ such that no sequence in $A$ converges to a point in $B,$ and no sequence in $B$ converges to a point in $A.$
The set $S$ is connected if no such partition exists. I.e., for any partition of $S$ into two disjoint nonempty subsets $A$ and $B,$ either there is a sequence in $A$ converging to a point in $B,$ or else there is a sequence in $B$ converging to a point in $A.$
A: 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.
A: The definition is what it is. There is no "more" or "less" trivial definition, since they are all equivalent.

That said, you may want a more intuitive explanation of connectedness.
Your definition is for example equivalent to saying that $E$ is connected if there do not exist two open sets $U,V$ such that $U\cap V=\emptyset$ and $U\cup V=E$.
The explanation is that intuitively,

A metric space is connected if it cannot be split into two pieces without "tearing" it somewhere.

A: A space $X$ is disconnected if there is a continuous function $f \colon X \to \{0,1\}$ that is not constant.
For example: $\mathbb R$ is connected because every map $f \colon \mathbb R \to \{0,1\}$ is constant (intermediate value theorem). The space $[0,1] \cup [2,3]$ is disconnected, because the function can be defined piecewise.
