# Category:Upper Closures

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This category contains results about Upper Closures in the context of Order Theory.

Definitions specific to this category can be found in Definitions/Upper Closures.

Let $\left({S, \preccurlyeq}\right)$ be an ordered set.

Let $a \in S$.

The **upper closure of $a$ (in $S$)** is defined as:

- $a^\succcurlyeq := \left\{{b \in S: a \preccurlyeq b}\right\}$

That is, $a^\succcurlyeq$ is the set of all elements of $S$ that succeed $a$.

## Pages in category "Upper Closures"

The following 24 pages are in this category, out of 24 total.

### I

- Infimum of Image of Upper Closure of Element under Increasing Mapping
- Infimum of Intersection of Upper Closures equals Join Operands
- Infimum of Upper Closure of Element
- Infimum of Upper Closure of Set
- Intersection of Upper Closures is Upper Closure of Join Operands
- Inverse Image under Order Embedding of Strict Upper Closure of Image of Point

### S

### U

- Upper and Lower Closures are Convex
- Upper Closure in Ordered Subset is Intersection of Subset and Upper Closure
- Upper Closure is Closure Operator
- Upper Closure is Decreasing
- Upper Closure is Smallest Containing Upper Set
- Upper Closure is Upper Set
- Upper Closure of Coarser Subset is Subset of Upper Closure
- Upper Closure of Element is Filter
- Upper Closure of Singleton
- Upper Closure of Subset is Subset of Upper Closure