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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?

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@5xum Good catch! Edit made – Cole Butler Mar 3 at 8:56
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A set is disconnected if it can be split into two disjoint nonempty subsets such that neither contains a limit point of the other. (Since we're talking about metric spaces, that just means that a sequence in one of the subsets can't converge to a point in the other one.) A set is connected if it cannot be split in such a way. – bof Mar 3 at 8:58
    
@bof You are correct! And thank you for the clarification! :) – Cole Butler Mar 3 at 9:02
up vote 12 down vote accepted

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.

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The set of nonnegative integers can be described as either $\Bbb N=\{x\in\Bbb Z\mid x\ge0\}$ or $\Bbb N=\{a^2+b^2+c^2+d^2\mid a,b,c,d\in\Bbb Z\}$, but I think you will agree that the first definition is "more trivial" / "simpler". Just because two definitions are equivalent doesn't mean they have the same complexity, and certainly doesn't mean the equivalence is simple. – Mario Carneiro Mar 3 at 19:27

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.

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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 $A.$

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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.

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"... it is not the same as path-connectedness", and the topologist's sine curve is an example of the difference. ​ ​ – Ricky Demer Mar 3 at 17:55
    
I do not understand the last sentence. – Carsten S Mar 3 at 21:26

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.

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