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Pseudo-intersection
In mathematical set theory, a pseudo-intersection of a family of sets is an infinite set ''S'' such that each element of the family contains all but a finite number of elements of ''S''. The pseudo-intersection number, sometimes denoted by the fraktur Fraktur () is a calligraphic hand of the Latin alphabet and any of several blackletter typefaces derived from this hand. It is designed such that the beginnings and ends of the individual strokes that make up each letter will be clearly vis ... letter đť”, is the smallest size of a family of infinite subsets of the natural numbers that has the strong finite intersection property but has no pseudo-intersection. References * Set theory {{settheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Properties The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC), but only by showing that it follows from the existence of the natural numbers. A set is infinite if and only if for every natural number, the set has a subset whose cardinality is that natural number. If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset. If a set of sets is infinite or contains an infinite element, then its union is infinite. The power set of an infinite set is infinite. Any superset of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fraktur
Fraktur () is a calligraphic hand of the Latin alphabet and any of several blackletter typefaces derived from this hand. It is designed such that the beginnings and ends of the individual strokes that make up each letter will be clearly visible, and often emphasized; in this way it is often contrasted with the curves of the Antiqua (common) typefaces where the letters are designed to flow and strokes connect together in a continuous fashion. The word "Fraktur" derives from Latin ("a break"), built from , passive participle of ("to break"), which is also the root for the English word "fracture". In non-professional contexts, the term "Fraktur" is sometimes misused to refer to ''all'' blackletter typefaces while Fraktur typefaces do fall under that category, not all blackletter typefaces exhibit the Fraktur characteristics described above. Fraktur is often characterized as "the German typeface", as it remained popular in Germany and much of Eastern Europe far longer than el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Finite Intersection Property
In general topology, a branch of mathematics, a non-empty family A of subsets of a set X is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of A is non-empty. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of A 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. Other applications include proving that certain perfect sets are uncountable, and the construction of ultrafilters. Definition Let X be a set and \mathcal a nonempty family of subsets of that is, \mathcal is a nonempty subset of the power set of Then \mathcal is said to have the finite intersection property if every nonempty finite subfamily has nonempty intersection; it is said to have the strong f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
