In Principles of Mathematical Analysis, why does Rudin define connectedness by separated sets?

Question

In Principles of Mathematical Analysis, why does Rudin define connectedness by separated sets? It seems to me that the standard definition with partitioning into two disjoint open sets seems much simpler.

For reference, here is Rudin's definition:

2.45   Definition   Two subsets $A$ and $B$ of a metric space $X$ are said to be separated if both $A \cap \bar{B}$ and $\bar{A} \cap B$ are empty, i.e., if no point of $A$ lies in the closure of $B$ and no point of $B$ lies in the closure of $A$.
     A set $E \subset X$ is said to be connected if $E$ is not a union of two nonempty separated sets.

Answer 1

It looks to me that the advantage is that the "standard" definition as you call it, requires looking at the subspace topology (as $E$ need not be open). Rudin's definition avoids that and defines connectedness using only the metric.

The closure is needed to avoid situations like partitioning $[0,2]$ in $[0,1]$ and $(1,2]$ or for a more dramatic example, $$ (\{0\}\times[0,1] )\cup\left\{\sin\frac1x:\ x\in(0,1]\right\}\subset\mathbb R^2. $$

Answer 2

Chapter 2 of Rudin's book defines metric spaces, not topological spaces. Right after he defines a metric space, he states:

It is important to observe that every subset $Y$ of a metric space $X$ is a metric space in its own right, with the same distance function.

He can't easily port over the standard (simpler but abstract) definition of being a connected topological space to his metric space setting. Indeed, by the time he explained it all you might then ask why he doesn't make Chapter 2 an introduction to general topology.

The cool thing about the definition of two separated sets $A$ and $B$ in a metric space $X$ is it immediately evident that if the two subsets are also contained in (metric) space $Y \subset X$ then they remain separated. So the notion of being a (Rudin) connected metric space $E$ does not depend on the ambient metric space $Z$ that it might be contained in.

Interestingly, the definitions that Rudin uses for separated sets and connected spaces works in the general topology framework; see #5 in this wikipedia passage.