---
tag: WY9BHD6GBA
kind: concept
name: Proximity space
msc: ["54E05"]
sources:
  - url: https://en.wikipedia.org/wiki/Proximity_space
same_as:
  wikidata: Q7252877
  wikipedia: Proximity space
---

## Summary

A proximity space is a set with a relation that says when two subsets are near each other.
Every metric space gives a proximity, in which two sets are near when their distance is $0$.
A proximity determines a topology, and it lies between a topology and a [[uniform space|A54T86KS63]] in the amount of structure it carries.

## Explanations

### L1 human

A topology tells when a point is near a set: when it lies in the closure of the set.
A proximity tells when two sets are near each other.
In the plane, the open disc of radius $1$ and the point $(1, 0)$ are near, since their distance is $0$, though they do not meet.
The open disc and the point $(2, 0)$ are not near.