What is defintion of partition of interval [a,b]?

Question

The book defines definition as

a partition of an interval [a, b] on the real line is a finite sequence of the form

a = x0 < x1 < x2 < ... < xn = b.

So does this basically mean that a partition of interval [a,b] is a subinterval within [a,b]?

in others words, does it mean an interval within an interval?

Answer 1

In mathematics, a partition of an interval [a, b] on the real line is a finite sequence of the form
a = x0 < x1 < x2 < ... < xn = b.
Such partitions are used in the theory of the Riemann integral, the Riemann-Stieltjes integral and the regulated integral.

The norm (or mesh) of the partition
x0 < x1 < x2 < ... < xn
is the length of the longest of these subintervals; it is
max{ |xi − xi−1| : i = 1, ..., n }.
As the mesh approaches zero, a Riemann sum based on the partition approaches the Riemann integral.
A tagged partition is a partition of an interval together with a finite sequence of numbers t0, ..., tn−1 subject to the conditions that for each i,
xi ti xi+1.

In other words, it is a partition together with a distinguished point of every subinterval. The mesh of a tagged partition is defined the same as for an ordinary partition. We can define a partial order on the set of all tagged partitions by saying that one tagged partition is bigger than another if the bigger one is a refinement of the smaller one.
Suppose that together with are a tagged partition of [a,b], and that together with are another tagged partition of [a,b]. We say that and together are a refinement of together with if for each integer i with , there is an integer r(i) such that xi = yr(i) and such that ti = sj for some j with . Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.

hope this helps

Answer 2

It has the same start and end point of [a,b], but not necessarilly all of the elements of [a,b] so it is a subinterval.