---
tag: ETCE287E2R
kind: concept
name: Upper set
msc: [06A06]
sources:
  - url: https://en.wikipedia.org/wiki/Upper_and_lower_sets
same_as:
  wikidata: Q2528262
  wikipedia: Upper and lower sets
---

## Summary

An upper set of a [[partially ordered set|011C29Q1S1]] is a subset that contains everything above each of its elements.
The dual notion is a lower set, which contains everything below each of its elements.
Upper and lower sets are used to define order filters, the Alexandrov topology, and the lattice of down-sets of a poset.

## Explanations

### L1 human

An upper set is closed upwards: once an element is in, every larger element is in too.
In the real numbers, the interval $[3, \infty)$ is an upper set, and so is $(3, \infty)$.
In the subsets of $\set{1, 2, 3}$ ordered by inclusion, the family of all subsets containing $1$ is an upper set.
The set $\set{3}$ alone is not an upper set of $\R$, because $4 > 3$ is missing.