In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X) of a partially ordered set ( X , ≤ ) {\displaystyle...

mathematics, particularly in order theory, an upper bound or majorant of a subset S of some preordered set (K, ≤) is an element of K that is greater than...

element is a lower bound of P. In our example, the set { x , y } {\displaystyle \{x,y\}} is an upper bound for the collection of elements { { x } , { y...

only if every upper set is also open (or equivalently every lower set is also closed). Let A be a subset of X. The smallest upper set containing A is...

Equivalently, an Alexandrov topology is one whose open sets are the upper sets for some preorder on the space. Spaces with an Alexandrov topology are...

a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in...

Formally, the upper closure of a set S in a poset P is given by the set {x in P | there is some y in S with y ≤ x}. A set that is equal to its upper closure...

than any other lower bound). Similarly, an upper bound of a subset S {\displaystyle S} of a partially ordered set ( P , ≤ ) {\displaystyle (P,\leq )} is an...

mathematics, a directed set (or a directed preorder or a filtered set) is a preordered set in which every finite subset has an upper bound. In other words...

The trading card game Magic: The Gathering has released a large number of sets since it was first published by Wizards of the Coast. After the 1993 release...

Lower Egypt, and ruled the Nile Delta, from Avaris. They chose Set, originally Upper Egypt's chief god, the god of foreigners and the god they found...

non-empty totally ordered set with no upper bound. The integers form an initial non-empty totally ordered set with neither an upper nor a lower bound. The...

For a preordered set P, any upper set O is Alexandrov-open. Inversely, a topology is Alexandrov if any intersection of open sets is open. Algebraic...

The Upper Deck Company, LLC (colloquially as Upper Deck and Upper Deck Authenticated, Ltd. in the UK) is a private company primarily known for producing...

{\displaystyle X} x ∈ X {\displaystyle x\in X} The upper contour set of x {\displaystyle x} is the set of all y {\displaystyle y} that are related to x...

In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and...

subset S of F is a base or basis for F if the upper set generated by S (i.e., the smallest upwards-closed set containing S) is equal to F. Since every filter...

sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets...

related types, through the difference in spans between its wings – the upper set being of much greater span than the lower. In 1926, the Ministry of National...

terms of a pair of sets which give the lower and the upper approximation of the original set. In the standard version of rough set theory described in...

The Upper Peninsula of Michigan—also known as Upper Michigan or colloquially the U.P. or Yoop—is the northern and more elevated of the two major landmasses...

Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally...

well-ordered set S contains for every subset T with an upper bound a least upper bound, namely the least element of the subset of all upper bounds of T...

Upper-atmospheric lightning and ionospheric lightning are terms sometimes used by researchers to refer to a family of short-lived electrical-breakdown...

set (upwards and downwards) centered and σ-centered set Net (mathematics) Upper set and lower set Ideal and filter Ultrafilter Completeness (order theory)...

partially ordered set, then its upper sets (the subsets of X with the additional property that if x belongs to an upper set U and x ≤ y, then y must also...

P is called Scott-open if it is an upper set and if it is inaccessible by directed joins, i.e. if all directed sets D with supremum in O have non-empty...

Upper class in modern societies is the social class composed of people who hold the highest social status. Usually, these are the wealthiest members of...

drive train is reconfigured to connect to the upper set of pedals, and the controls are moved to the upper handlebar area. Alternatively, a bicycle can...

A somewhat weaker notion of density is the upper Banach density d ∗ ( A ) {\displaystyle d^{*}(A)} of a set A ⊆ N . {\displaystyle A\subseteq \mathbb {N}...