Inmathematics, anupper set (also called anupward closed set, anupset, or anisotone set inX)^[1] of apartially ordered set${\displaystyle (X,\le )}$ is a subset${\displaystyle S\subseteq X}$ with the following property: ifs is inS and ifx inX is larger thans (that is, if), thenx is inS. In other words, this means that anyx element ofX that is${\displaystyle \ge }$ to some element ofS is necessarily also an element ofS. The termlower set (also called adownward closed set,down set,decreasing set,initial segment, orsemi-ideal) is defined similarly as being a subsetS ofX with the property that any elementx ofX that is${\displaystyle \le }$ to some element ofS is necessarily also an element ofS.

Let${\displaystyle (X,\le )}$ be apreordered set. Anupper set in${\displaystyle X}$ (also called anupward closed set, anupset, or anisotone set)^[1] is a subset${\displaystyle U\subseteq X}$ that is "closed under going up", in the sense that

for all${\displaystyle u\in U}$ and all${\displaystyle x\in X,}$ if${\displaystyle u\le x}$ then${\displaystyle x\in U.}$

Thedual notion is alower set (also called adownward closed set,down set,decreasing set,initial segment, orsemi-ideal), which is a subset${\displaystyle L\subseteq X}$ that is "closed under going down", in the sense that

for all${\displaystyle l\in L}$ and all${\displaystyle x\in X,}$ if${\displaystyle x\le l}$ then${\displaystyle x\in L.}$

The termsorder ideal orideal are sometimes used as synonyms for lower set.^[2][3][4] This choice of terminology fails to reflect the notion of an ideal of alattice because a lower set of a lattice is not necessarily a sublattice.^[2]

• Every partially ordered set is an upper set of itself.
• Theintersection and theunion of any family of upper sets is again an upper set.
• Thecomplement of any upper set is a lower set, and vice versa.
• Given a partially ordered set${\displaystyle (X,\le ),}$ the family of upper sets of${\displaystyle X}$ ordered with theinclusion relation is acomplete lattice, theupper set lattice.
• Given an arbitrary subset${\displaystyle Y}$ of a partially ordered set${\displaystyle X,}$ the smallest upper set containing${\displaystyle Y}$ is denoted using an up arrow as${\displaystyle \uparrow Y}$ (seeupper closure and lower closure).
  • Dually, the smallest lower set containing${\displaystyle Y}$ is denoted using a down arrow as${\displaystyle \downarrow Y.}$
• A lower set is calledprincipal if it is of the form${\displaystyle \downarrow \{x\}}$ where${\displaystyle x}$ is an element of${\displaystyle X.}$
• Every lower set${\displaystyle Y}$ of a finite partially ordered set${\displaystyle X}$ is equal to the smallest lower set containing allmaximal elements of${\displaystyle Y}$
  • ${\displaystyle \downarrow Y=\downarrow \mathrm{Max}(Y)}$ where${\displaystyle \mathrm{Max}(Y)}$ denotes the set containing the maximal elements of${\displaystyle Y.}$
• Adirected lower set is called anorder ideal.
• For partial orders satisfying thedescending chain condition, antichains and upper sets are in one-to-one correspondence via the followingbijections: map each antichain to its upper closure (see below); conversely, map each upper set to the set of its minimal elements. This correspondence does not hold for more general partial orders; for example the sets ofreal numbers${\displaystyle \{x\in \mathbb{R}:x>0\}}$ and${\displaystyle \{x\in \mathbb{R}:x>1\}}$ are both mapped to the empty antichain.

Given an element${\displaystyle x}$ of a partially ordered set${\displaystyle (X,\le ),}$ theupper closure orupward closure of${\displaystyle x,}$ denoted by${\displaystyle x^{\uparrow X},}$${\displaystyle x^{\uparrow },}$ or${\displaystyle \uparrow x,}$ is defined by$${\displaystyle x^{\uparrow X}=\uparrow x=\{u\in X:x\le u\}}$$while thelower closure ordownward closure of${\displaystyle x}$, denoted by${\displaystyle x^{\downarrow X},}$${\displaystyle x^{\downarrow },}$ or${\displaystyle \downarrow x,}$ is defined by$${\displaystyle x^{\downarrow X}=\downarrow x=\{l\in X:l\le x\}.}$$

The sets${\displaystyle \uparrow x}$ and${\displaystyle \downarrow x}$ are, respectively, the smallest upper and lower sets containing${\displaystyle x}$ as an element. More generally, given a subset${\displaystyle A\subseteq X,}$ define theupper/upward closure and thelower/downward closure of${\displaystyle A,}$ denoted by${\displaystyle A^{\uparrow X}}$ and${\displaystyle A^{\downarrow X}}$ respectively, as$${\displaystyle A^{\uparrow X}=A^{\uparrow }=\bigcup _{a\in A}\uparrow a}$$and$${\displaystyle A^{\downarrow X}=A^{\downarrow }=\bigcup _{a\in A}\downarrow a.}$$

In this way,${\displaystyle \uparrow x=\uparrow \{x\}}$ and${\displaystyle \downarrow x=\downarrow \{x\},}$ where upper sets and lower sets of this form are calledprincipal. The upper closure and lower closure of a set are, respectively, the smallest upper set and lower set containing it.

The upper and lower closures, when viewed as functions from the power set of${\displaystyle X}$ to itself, are examples ofclosure operators since they satisfy all of theKuratowski closure axioms. As a result, the upper closure of a set is equal to the intersection of all upper sets containing it, and similarly for lower sets. (Indeed, this is a general phenomenon of closure operators. For example, thetopological closure of a set is the intersection of allclosed sets containing it; thespan of a set of vectors is the intersection of allsubspaces containing it; thesubgroup generated by a subset of agroup is the intersection of all subgroups containing it; theideal generated by a subset of aring is the intersection of all ideals containing it; and so on.)

Anordinal number is usually identified with the set of all smaller ordinal numbers. Thus each ordinal number forms a lower set in the class of all ordinal numbers, which are totally ordered by set inclusion.

• Abstract simplicial complex (also called:Independence system) - a set-family that is downwards-closed with respect to the containment relation.
• Cofinal set – a subset${\displaystyle U}$ of a partially ordered set${\displaystyle (X,\le )}$ that contains for every element${\displaystyle x\in X,}$ some element${\displaystyle y}$ such that${\displaystyle x\le y.}$

• Blanck, J. (2000)."Domain representations of topological spaces"(PDF).Theoretical Computer Science.247 (1–2):229–255.doi:10.1016/s0304-3975(99)00045-6.
• Dolecki, Szymon; Mynard, Frédéric (2016).Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company.ISBN 978-981-4571-52-4.OCLC 945169917.
• Hoffman, K. H. (2001),The low separation axioms (T₀) and (T₁)

• Order theory

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