nLab Richman premetric space

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

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Idea

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

Definition

A Richman premetric space is a set $S$ with a ternary relation $(-)\sim_{(-)}(-)\colon S \times \mathbb{Q}_{\geq 0} \times S \to \Omega$, where $\mathbb{Q}_{\geq 0}$ represent the non-negative rational numbers in $\mathbb{Q}$ and $\Omega$ is the set of truth values, such that

• for all $x \in S$ and $y \in S$, $(x = y) \iff (x \sim_0 y)$

• for all $x \in S$ and $y \in S$, there exists $q \in \mathbb{Q}_{\geq 0}$ such that $x \sim_q y$

• for all $x \in S$, $y \in S$, $q \in \mathbb{Q}_{\geq 0}$, and $r \in (q, \infty)$, where $(q, \infty)$ is the set of all non-negative rational numbers strictly greater than $q$, then $(x \sim_r y) \iff (x \sim_q y)$

• for all $x \in S$, $y \in S$, $z \in S$, $q \in \mathbb{Q}_{\geq 0}$, and $r \in \mathbb{Q}_{\geq 0}$, if $x \sim_q y$ and $y \sim_r z$, then $x \sim_{q + r} z$.

Properties

Assuming excluded middle, every Richman premetric space is a metric space. Without excluded middle, however, every Richman premetric space is a “metric space” which is valued in the lower Dedekind real numbers, rather than the two-sided Dedekind real numbers.

See also

• premetric space

• Booij premetric space

• metric space

References

• Fred Richman, Real numbers and other completions, Mathematical Logic Quarterly 54 1 (2008) 98-108 [doi:10.1002/malq.200710024]