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(0,1)-category

(0,1)-topos

# Contents

## Definition

In a poset or even proset, a lower set $L$ is a subset that is ‘downward closed’; that is,

• whenever $y \leq x$ and $x \in L$, then $y \in L$.

Lower sets form a Moore collection and so one can speak of the lower set generated by an arbitrary subset $A$:

$A{\downarrow} = \{ y \;|\; \exists x,\; x \in A \;\wedge\; y \leq x \} .$

Sometimes a lower set is called an ‘ideal’, but that term can also mean something more specific (and always does in a lattice).

A lower set is also sometimes called a ‘down set’, but that term can also mean something more specific: the down set of $x$ is the lower set generated by $x$.

The characteristic function of a lower set is precisely a (0,1)-presheaf.

## Examples

An inhabited, open lower set of rational numbers (or equivalently of real numbers) determines precisely a lower real number.

## See also

Last revised on July 22, 2022 at 08:26:47. See the history of this page for a list of all contributions to it.