A partition of an inteval $[a,b]$ of $\mathbb{R}$ is generally defined as a finite sequence of the form:$$a = x_0 < x_1 < x_2 < \dots < x_n = b$$
Then, $[a,b]$ is seen as the following union of intervals: $[x_0,x_1] \cup [x_1,x_2]\cup\dots\cup[x_{n-1},x_n]$.
I wonder if it makes sense to extend this definition to sequences of the form:$$a = x_0 \leq x_1 \leq x_2 \leq \dots \leq x_n = b$$
Then, degenerate intervals (i.e. intervals with equal bounds) are allowed.
For instance, if $a=x_0=x_1=x_2=...=x_{n-1}<x_n=b$ then$$[a,b] = [x_0,x_1] \cup [x_1,x_2]\cup\dots\cup[x_{n-1},x_n] = [x_0,x_0] \cup [x_0,x_1]\cup...\cup [x_0,x_n]$$
It this still a partition?
