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...
9 KB (1,290 words) - 14:45, 19 June 2025
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...
6 KB (687 words) - 21:09, 17 June 2025
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...
40 KB (5,351 words) - 15:10, 28 June 2025
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...
10 KB (1,414 words) - 15:58, 2 May 2025
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...
12 KB (1,604 words) - 13:26, 8 July 2025
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...
12 KB (1,465 words) - 01:39, 2 July 2025
Order theory (section Partially ordered sets)
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...
31 KB (4,490 words) - 06:40, 21 June 2025
Infimum and supremum (redirect from Least upper bound)
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...
25 KB (4,523 words) - 14:12, 31 December 2024
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...
16 KB (2,675 words) - 15:44, 14 June 2025
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...
231 KB (11,901 words) - 00:59, 7 July 2025
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...
37 KB (4,579 words) - 03:49, 1 July 2025
Total order (redirect from Total ordered set)
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...
22 KB (3,100 words) - 10:51, 4 June 2025
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...
29 KB (4,204 words) - 03:05, 12 April 2025
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...
57 KB (6,533 words) - 23:43, 5 May 2025
{\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...
7 KB (1,288 words) - 19:59, 9 December 2021
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...
5 KB (901 words) - 05:00, 6 June 2025
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...
24 KB (2,809 words) - 11:02, 7 July 2025
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...
49 KB (7,143 words) - 21:00, 7 July 2025
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...
9 KB (865 words) - 22:29, 9 July 2025
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...
52 KB (8,170 words) - 20:42, 10 June 2025
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...
138 KB (10,420 words) - 17:27, 1 July 2025
Zorn's lemma (category Lemmas in set theory)
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...
32 KB (4,665 words) - 21:22, 30 June 2025
Well-order (redirect from Well-ordered set)
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...
12 KB (1,902 words) - 19:06, 15 May 2025
Upper-atmospheric lightning and ionospheric lightning are terms sometimes used by researchers to refer to a family of short-lived electrical-breakdown...
33 KB (3,556 words) - 11:43, 6 July 2025
set (upwards and downwards) centered and σ-centered set Net (mathematics) Upper set and lower set Ideal and filter Ultrafilter Completeness (order theory)...
5 KB (396 words) - 23:32, 16 April 2025
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...
8 KB (1,424 words) - 05:09, 22 April 2024
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...
6 KB (753 words) - 00:02, 14 May 2025
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...
23 KB (2,273 words) - 20:01, 18 June 2025
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...
7 KB (848 words) - 09:40, 4 June 2025
Natural density (redirect from Set density)
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}...
11 KB (1,762 words) - 17:14, 12 June 2025