Filters in topologyFilters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such as convergence, continuity, compactness, and more. Filters, which are special families of subsets of some given set, also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called have many useful technical properties and they may often be used in place of arbitrary filters.
Duality (order theory)In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the same set, but with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P. It is easy to see that this construction, which can be depicted by flipping the Hasse diagram for P upside down, will indeed yield a partially ordered set. In a broader sense, two partially ordered sets are also said to be duals if they are dually isomorphic, i.
Join and meetIn mathematics, specifically order theory, the join of a subset of a partially ordered set is the supremum (least upper bound) of denoted and similarly, the meet of is the infimum (greatest lower bound), denoted In general, the join and meet of a subset of a partially ordered set need not exist. Join and meet are dual to one another with respect to order inversion. A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice.
PréordreEn mathématiques, un préordre est une relation binaire réflexive et transitive. C'est-à-dire que si E est un ensemble, une relation binaire sur E est un préordre lorsque : (réflexivité) ; (transitivité). Un ensemble préordonné est un ensemble muni d'un préordre, ou plus formellement un couple où désigne un ensemble et un préordre sur . Les ordres sont les préordres antisymétriques. Les relations d'équivalence sont les préordres symétriques. Dans un anneau commutatif, la relation « divise » est une relation de préordre.
Finite intersection propertyIn general topology, a branch of mathematics, a non-empty family A of subsets of a set is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of is non-empty. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of is infinite. Sets with the finite intersection property are also called centered systems and filter subbases. The finite intersection property can be used to reformulate topological compactness in terms of closed sets; this is its most prominent application.
Ensemble ordonné filtrantEn mathématiques, un ensemble ordonné filtrant est un ensemble ordonné (c'est-à-dire dans lequel on peut dire que certains éléments sont plus grands que d'autres) tel que pour toute paire d'éléments, il existe un élément qui est plus grand que chaque élément de la paire. Cela sous-entend en premier lieu que ce troisième élément peut être comparé aux deux premiers, ce qui n'est pas automatique dans un ensemble ordonné (implicitement partiellement ordonné, par opposition à totalement ordonné).
Section commençanteEn mathématiques, et plus précisément en théorie des ordres, une section commençante (également appelée segment initial ou sous-ensemble fermé inférieurement) d'un ensemble ordonné (X,≤) est un sous-ensemble S de X tel que si x est dans S et si y ≤ x, alors y est dans S. Dualement, on appelle section finissante (ou sous-ensemble fermé supérieurement) un sous-ensemble F tel que si x est dans F et si x ≤ y, alors y est dans F.
