# Definition of connected sets

Wikipedia provides several different definitions of connectedness that they say are equivalent. For example, it says that a connected set is not the union of disjoint open sets. While this makes sense, why does it suffice to show that if a set is not the union of open sets, it is connected? (Ex. The union of [1,2] and [3,4] is not connected, but it is not the union of open sets.)

• Those sets are open in the correct subspace topology. Commented May 22, 2020 at 19:17
• If the Wikipedia page didn't say "nonempty open sets" then it needs to be corrected.
– bof
Commented Jul 4, 2020 at 7:04

$$[1,2]\cup[3,4]$$ is union of two open sets and so it is disconnected. Important is looking carefully the definition.

Definition:

A metric space $$X$$ is said to disconnected if it is union of two non empty disjoint open sets.
A subset $$S\subset X$$ is said to be disconnected if it is disconnected as a subspace of $$X$$.

So $$S$$ will become disconnected if $$S=U\cup V$$ for disjoint nonempty open sets $$U$$ and $$V$$ in $$S$$.

Now here if you talk about connectedness of $$S=[1,2]\cup[3,4]$$ then we have to consider it as a subspace of $$\Bbb{R}$$. Here $$[1,2]=S\cap\left(\displaystyle\frac12,\frac52\right)$$ and

$$[3,4]= S\cap\left(\displaystyle\frac52,\frac92\right)$$ are open sets in $$S$$. Thus $$S$$ is union of two non empty disjoint open sets and hence not connected.

We have to understand that connectedness must be defined in a metric (or topological) space. So, when we say "open" sets, we mean that they are open subject to a defined (metric) topology.

Simply, the connected (space) set $$A \subseteq X$$ is a set that is contained in a metric (or topological) space and there are not any two disjoint open (which implies the closed too) sets (in $$X$$) that make a partition for $$A$$. Namely, for any $$U$$ and $$V$$ non-empty open in $$X$$ and $$U \cap V =\phi$$, then $$U \cup V \neq A$$.

In conclusion, a set $$A$$ is connected if we can not find a partition for A from the open sets (subject to the topology (metric)).

• Somewhere in the sentence beginning "Namely" you need to say that the open sets are nonempty. (You were okay in the preceding sentence because "nonempty" is implicit in "partition".)
– bof
Commented Jul 4, 2020 at 7:03

What is meant is that the sets should be open in the space itself. In the union of $$[1,2]$$ and $$[3,4]$$, each of the intervals is open in the subspace topology, even though they are not open in the ambient topology.

A disconnected space is one that is exactly equal to the disjoint union of two of its nonempty open subsets. For a subspace $$A$$, it's enough for there to exist two nonempty open sets $$U$$ and $$V$$ in the ambient space such that $$U\cap A$$ and $$V\cap A$$ are disjoint and $$A\subseteq U\cup V$$.

• @bof True, fixed. Commented Jul 4, 2020 at 11:45