nLab Booij premetric space

This entry is about the family of binary relations indexed by the positive rational numbers defined by Auke Booij in Analysis in univalent type theory. For the family of binary relations indexed by the non-negative rational numbers defined by Fred Richman in Real numbers and other completions, see Richman premetric space.

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Idea

A more general concept of metric space by Auke Booij. While Auke Booij simply called these structures “premetric spaces”, there are multiple notions of premetric spaces in the mathematical literature, so we shall refer to these as Booij premetric spaces.

Definition

A Booij premetric space is a set $S$ with a ternary relation $a \sim_\epsilon b$ for $a \in S$, $b \in S$, and $\epsilon \in \mathbb{Q}_+$, where $\mathbb{Q}_+$ represent the positive rational numbers in $\mathbb{Q}$.

Booij premetric spaces are used as in the construction of the Cauchy real numbers and the HoTT book real numbers.

Uniform Booij premetric spaces

A Booij premetric space is uniform if

1. for every positive rational number $\epsilon \in \mathbb{Q}_+$ and $a \in S$, $a \sim_\epsilon a$

2. for every positive rational number $\epsilon \in \mathbb{Q}_+$, there exists a positive rational number $\delta \in \mathbb{Q}_+$ such that for every $a \in S$, $b \in S$, and $c \in S$, if $a \sim_\epsilon b$ and $b \sim_\epsilon c$, then $a \sim_\delta c$.

3. For every positive rational number $\epsilon \in \mathbb{Q}_+$, there exists a positive rational number $\delta \in \mathbb{Q}_+$ such that for every $a \in S$ and $b \in S$, if $a \sim_\epsilon b$, then $b \sim_\delta a$.

4. There is a positive rational number $\epsilon \in \mathbb{Q}_+$ and elements $a \in S$ and $b \in S$ such that $a \sim_\epsilon b$.

5. If there are positive rational numbers $\epsilon \in \mathbb{Q}_+$ and $\delta \in \mathbb{Q}_+$, then there is a positive rational number $\eta \in \mathbb{Q}_+$ such that for every element $a \in S$ and $b \in S$, if $a \sim_\eta b$, then $a \sim_\epsilon b$ and $a \sim_\delta b$.

6. If there is a positive rational number $\epsilon \in \mathbb{Q}_+$, then there is a positive rational number $\delta \in \mathbb{Q}_+$ such that for all $a \in S$ and $b \in S$, if $a \sim_\epsilon b$, then $a \sim_\delta b$.

Generalizations

Booij premetric spaces could be generalized from the positive rational numbers $\mathbb{Q}_+$ to any set $T$. The binary relation $\sim_U$ for each element $U \in T$ is called an entourage. These are called $T$-premetric spaces and are sets $S$ with a ternary relation $a \sim_U b$ for $a \in S$, $b \in S$, and $U \in T$. These could also be used to construct the HoTT book real numbers if one only has an integral subdomain $R \subseteq \mathbb{Q}$ such as the dyadic rational numbers or the decimal rational numbers, where the index set $T$ is $R_+$ rather than $\mathbb{Q}_+$.

Uniform spaces

Seven axioms are usually added to a $T$-premetric spaces, to get different variants of a uniform space.

1. for every element $U \in T$ and $a \in S$, $a \sim_U a$

2. for every element $U \in T$, there exists an element $V \in T$ such that for every $a \in S$, $b \in S$, and $c \in S$, if $a \sim_V b$ and $b \sim_V c$, then $a \sim_U c$.

3. For every element $U \in T$, there exists an element $V \in T$ such that for every $a \in S$ and $b \in S$, if $a \sim_V b$, then $b \sim_U a$.

4. There is an element $U \in T$ and elements $a \in S$ and $b \in S$ such that $a \sim_U b$.

5. If there are element $U \in T$ and $V \in T$, then there is an element $W \in T$ such that for every element $a \in S$ and $b \in S$, if $a \sim_W b$, then $a \sim_U b$ and $a \sim_V b$.

6. If there is an element $U \in T$, then there is an element $V \in T$ such that for all $a \in S$ and $b \in S$, if $a \sim_U b$, then $a \sim_V b$.

7. For every element $U \in T$, there exists an element $V \in T$ such that for all elements $a \in S$ and $b \in S$, $a \sim_U b$ or $\neg (a \sim_V b)$.

Examples of these structures include

• uniformly locally decomposable spaces satisfy all 7 axioms.

• uniform spaces satisfy axioms 1-6

• quasiuniformly locally decomposable spaces satisfy axioms 1, 2, and 4-7

• quasiuniform spaces satisfy axioms 1, 2, and 4-6

• preuniform spaces satisfy axioms 1-3

• quasipreuniform spaces? satisfy axioms 1 and 2

 Euclidean spaces

Let $R$ be an ordered integral domain, and let $V$ be a finite rank $R$-module with basis $X:\mathrm{Fin}(n) \to V$ and positive definite bilinear inner product $\langle -, -\rangle:V \times V \to R$, where $\langle X_i, X_j \rangle = \delta_i^j$ and $\delta_{(-)}^{(-)}:\mathrm{Fin}(n) \times \mathrm{Fin}(n) \to R$ is the Kronecker delta indexed by $\mathrm{Fin}(n)$.

We define the Euclidean premetric $a \sim_\epsilon b$ indexed by $a \in V$, $b \in V$, and $\epsilon \in R_+$ as

$$a \sim_\epsilon b \coloneqq \sum_{i = 0}^{n - 1} \langle a - b, X_i\rangle^2 \lt \epsilon^2$$

which makes $V$ an Euclidean space.

Convergence

Every $T$-premetric space is a generalised sequential space with the convergence relation $x \to b$ between sequences $x$ and elements $b$ true if and only if for all elements $U \in T$, there exists a natural number $N \in \mathbb{N}$ such that for all natural numbers $n \geq N$, $x_n \sim_U b$.

Given a $T$-premetric space $(S, \sim)$ and a sequence $x:\mathbb{N} \to S$, a modulus of Cauchy convergence is a function $M:T \to \mathbb{N}$ such that for all elements $U \in T$ and for all natural numbers $m \geq M(\epsilon)$ and $n \geq M(\epsilon)$, $x_m \sim_U x_n$.

A $T$-premetric space $S$ is sequentially Cauchy complete if every Cauchy sequence has a unique limit. A $T$-premetric space $S$ is sequentially modulated Cauchy complete if every sequence with a modulus of Cauchy convergence has a unique limit.

All this could be generalised to nets, since every $T$-premetric space is also a generalised net space with the convergence relation $x \to b$ between nets $x$ and elements $b$ true if and only if for all elements $U \in T$ and directed sets $I$, there exists an element $N \in I$ such that for all elements $n \geq N$, $x_n \sim_U b$.

See also

• premetric space

• Richman premetric space

• Cauchy structure

• metric space

References

• Auke B. Booij, Analysis in univalent type theory (pdf)