In many analysis courses and books there are two main definitions of connected sets that may not be the same (I know that in both cases a connected set is intended to be defined as a set that has no "holes"). Which one would you say that is a more complete definition?
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One definition may be: "A metric space $S$ is called disconnected if $S= A \cup B$, where $A$ and $B$ are disjoint nonempty open sets in $S$. We call $S$ connected if it is not disconnected."
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Another definition may be: "Two subsets $A$ and $B$ of a metric space $X$ are said to be separated is both $A \cap \bar{B}$ and $ \bar{A} \cap B$ are empty (where $\bar{A}$ and $\bar{B}$ represent the closure of $A$ and $B$). A set $E \subset X$ is said to be connected if $E$ is not a union of two nonempty separated sets."
As can be seen, the first definition requieres $A$ and $B$ to be open disjoint sets in a metric space while the second definition requires $A$ and $B$ to be separated which is not the same as open and disjoint, because closed disjoints sets are also separated. So it seems to me that the second definition is more general and maybe more complete. My question is, which is more useful for studying topology/analysis or mathematics in general?
Best Answer
In a topological space, a finite partition made from open sets is exactly the same as a finite partition made of closed sets... and this is just the same as a finite partition made of clopen sets.
Those two definitions are in fact different, because item 1 defines what a connected topological space is. While item 2 defines what a connected subset of a topological space is.
IMO, a better definition of a connected subset is a set that is a connected topological space in the relative topology. Whenever people define topological subset skipping the relative topology, they end up with some (usually ugly) trick to talk about it in some obscure fashion. In this case, notice that \begin{align*} A \cap \overline{B} = \emptyset &\Leftrightarrow E \cap \overline{B} = B \\&\Leftrightarrow \text{$B$ is closed in $E$ (with its relative topology)}. \end{align*} So, $E = A \cup B$, and both are (disjoint) closed (and therefore open... and therefore clopen) in $E$.
By the way, there are many different and interesting equivalent definitions for a connected topological space. Regardless of which one you choose, I think the best definition for a connected subset is the one that uses the relative topology.