Partition of a Set is defined as "A collection of disjoint subsets of a given set. The union of the subsets must equal the entire original set." For example, one possible partition of $(1, 2, 3, 4, 5, 6 )$ is $(1, 3), (2), (4, 5, 6).$ Rudin, while defining integral on page $120$ starts like this,
Definition Let $[a, b]$ be a given interval. By a partition $P$ of $[a,b]$ we mean a finite set of points $x_0, x_1,..., x_n$, where $a=x_0\leq x_1\leq...\leq x_n=b$.
if all the points from $a$ to $b$ are in partition $P$ then where is the other partition, is it $\phi$?, not mentioned in the book.
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$\begingroup$This is a slightly different kind of partition. Here the idea is that the interval $[a, b]$ is being partitioned into sub-intervals $[x_0, x_1], [x_1, x_2], \ldots$.
As with the kind of partition you defined, the sub-intervals here completely cover the original set $[a, b]$. Unlike with the kind of partition you defined, the sub-intervals here are not exactly disjoint. Instead they are almost disjoint, since they overlap only at their endpoints.
Rudin says that the points $x_0, x_1,\ldots$ "are" the partition, but that is just because once you know those points, you know everything there is to know about the way that $[a,b]$ has been divided into sub-intervals. In a more general setting, with the definition you quoted, that is not the case.
$\endgroup$ 2 $\begingroup$You're looking at two different definitions. The second definition is used to develop the theory of integration.
$\endgroup$ $\begingroup$The first definition of a partition is the one that is more generally used.
However, if the context of Rudin's book, he is likely trying to define the integral. This definition different. However, note that $[x_0, x_1]$, $(x_1, x_2]$, ..., $(x_{n-1}, x_n]$ is a partition in the first sense. However, Rudin's definition of partition does not account for all possible partition of $[a,b]$ in the first sense.
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