nLab premetric space (Richman)

This article is about premetric spaces as defined by Richman 2008. For other notions of premetric spaces, see 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.

Definition

A 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 premetric space is a metric space. Without excluded middle, however, every 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

• premetric space (Booij)

• metric space

References

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