nLab premetric space (Booij)

This article is about premetric spaces as defined by Booij 2020. For other notions of premetric spaces, see premetric space.

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Idea

A more general concept of metric space.

Definition

A 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 $\mathbb{Q}$.

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

Real premetric spaces

The usual notion of a metric space uses the real numbers rather than the rational numbers in the metric inequalities. This means that to generalize from metric spaces, one can use the positive real numbers instead of the positive rational numbers as the indexing set of the ternary relation (i.e. $a \sim_\epsilon b$ for $a \in S$, $b \in S$, and $\epsilon \in \mathbb{R}_+$), yielding a notion of a real premetric space. The original notion of a premetric space by Booij can then be called a rational premetric space.

Uniform premetric spaces

A rational or real premetric space is uniform if

1. for every positive number $\epsilon$ and $a \in S$, $a \sim_\epsilon a$

2. for every positive number $\epsilon$, there exists a positive number $\delta$ 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 number $\epsilon$, there exists a positive number $\delta$ 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 number $\epsilon$ and elements $a \in S$ and $b \in S$ such that $a \sim_\epsilon b$.

5. If there are positive numbers $\epsilon$ and $\delta$, then there is a positive number $\eta$ 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 number $\epsilon$, then there is a positive number $\delta$ such that for all $a \in S$ and $b \in S$, if $a \sim_\epsilon b$, then $a \sim_\delta b$.

Generalizations

Premetric spaces could be generalized from the positive rational or real numbers 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 complete if every Cauchy sequence has a unique limit. A $T$-premetric space $S$ is regularly sequentially 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

• premetric space (Gilbert)

• premetric space (Richman)

• Cauchy structure

• metric space

References

• Auke Booij, Analysis in Univalent Type Theory (2020) [etheses:10411, pdf, pdf]