The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $a,b\in\mathbb{R}$ with $a<b$, and let partitions of $[a,b]$ and their refinements be as in Partitions of a Closed Interval, the Set of Partitions, and Refinements. Let $[m]=\{1,\dots,m\}$ for $m\in\mathbb{N}_{0}$ be as in Intervals of Natural Numbers (ii), and let $\#P$ be the number of elements of a finite set $P$. Let $P$ be a partition of $[a,b]$.

(i) There is exactly one $n\in\mathbb{N}$ for which some bijection $x:[n+1]\to P$, $i\mapsto x_{i}$, satisfies $x_{i}<x_{j}$ for all $i,j\in[n+1]$ with $i<j$; for this $n$ there is exactly one such bijection $x$; and $\#P=n+1$, $x_{1}=a$ and $x_{n+1}=b$, so that $a=x_{1}<x_{2}<\dots<x_{n+1}=b$.

(ii) With $n$ and $x$ as in (i): for $i\in[n]$, both $i$ and $i+1$ lie in $[n+1]$, by Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting (i), Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting (ii) and Arithmetic and Order of the Natural Numbers (ix), so $x_{i}$ and $x_{i+1}$ are defined.

(iii) $x_{i+1}-x_{i}>0$ for every $i\in[n]$; the map $[n]\to\mathbb{R}$, $i\mapsto x_{i+1}-x_{i}$, exists by The Map of Consecutive Differences of a Finite List of Reals Exists, and Its Sum Telescopes (i); and, for the finite sum of Iterated Operations: Finite Sums and Finite Products (iii), which is also the sum over the finite set $[n]$ by Natural Numbers as Indices: ℕ ⊆ ℕ₀, 1 ∈ ℕ₀, and the Two Readings of an Operation from 1 to n (v),

$$\sum_{i=1}^{n}(x_{i+1}-x_{i})=b-a .$$

(iv) For every $t\in[a,b]$ there is $i\in[n]$ with $x_{i}\le t\le x_{i+1}$.

(v) If $Q$ is a partition of $[a,b]$, then $P\cup Q$ is a partition of $[a,b]$ and a refinement of both $P$ and $Q$.

(vi) Let $y\in[a,b]$. Then $P\cup\{y\}$ is a partition of $[a,b]$ and a refinement of $P$. If $y\notin P$, then there is exactly one $j\in[n]$ with $x_{j}<y<x_{j+1}$; for $P\cup\{y\}$ the $n$ of (i) is $n+1$, and the bijection $z:[(n+1)+1]\to P\cup\{y\}$ of (i) for $P\cup\{y\}$ satisfies $z_{k}=x_{k}$ for every $k\in[n+1]$ with $k\le j$, $z_{j+1}=y$, and $z_{k+1}=x_{k}$ for every $k\in[n+1]$ with $j<k$.

(vii) Let $c\in\mathbb{R}$ with $a<c<b$, let $P_{1}$ be a partition of $[a,c]$ and $P_{2}$ a partition of $[c,b]$, and let $p,q\in\mathbb{N}$ and $y:[p+1]\to P_{1}$, $z:[q+1]\to P_{2}$ be the $n$ and the bijection of (i) for $P_{1}$ and for $P_{2}$, respectively. Then $P_{1}\cup P_{2}$ is a partition of $[a,b]$; for it the $n$ of (i) is $p+q$, and its bijection $x:[(p+q)+1]\to P_{1}\cup P_{2}$ of (i) satisfies $x_{i}=y_{i}$ for every $i\in[p+1]$ and $x_{p+j}=z_{j}$ for every $j\in[q+1]$; here $[p+1]\subseteq[(p+q)+1]$ and $p+j\in[(p+q)+1]$ by Splitting the Segment [m+n] into [m] and a Shifted Copy of [n] (i), Splitting the Segment [m+n] into [m] and a Shifted Copy of [n] (ii), Arithmetic and Order of the Natural Numbers (i) and Arithmetic and Order of the Natural Numbers (ii).

(viii) Let $c\in\mathbb{R}$ with $a<c<b$. Then $(P\cup\{c\})\cap[a,c]$ is a partition of $[a,c]$, $(P\cup\{c\})\cap[c,b]$ is a partition of $[c,b]$, and their union is $P\cup\{c\}$.

(ix) Let $N\in\mathbb{N}$, the natural numbers being read in $\mathbb{R}$ as in The Real Line: Standing Notation and Background for Calculus on Intervals (vii), and let $h=(b-a)/N$, where $0<N$ in $\mathbb{R}$ by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (vi). For $k\in[N+1]$ let $w_{k}=a+(k-1)h$, where $1\le k$ by Arithmetic and Order of the Natural Numbers (x) and $k-1\in\mathbb{N}_{0}$ is the difference of The Difference of Two Natural Numbers with Zero (i). Then $\{w_{k}:k\in[N+1]\}$ is a partition of $[a,b]$; for it the $n$ of (i) is $N$ and the bijection of (i) is $k\mapsto w_{k}$; and $w_{k+1}-w_{k}=h$ for every $k\in[N]$, where $k,k+1\in[N+1]$ by (ii).