|
S And L Spaces
In mathematics, S-space is a regular topological space that is hereditarily separable but is not a Lindelöf space. L-space is a regular topological space that is hereditarily Lindelöf but not separable. A space is separable if it has a countable dense set and hereditarily separable if every subspace is separable. It had been believed for a long time that S-space problem and L-space problem are dual, i.e. if there is an S-space in some model of set theory then there is an L-space in the same model and vice versa – which is not true. It was shown in the early 1980s that the existence of S-space is independent of the usual axioms of ZFC. This means that to prove the existence of an S-space or to prove the non-existence of S-space, we need to assume axioms beyond those of ZFC. The L-space problem (whether an L-space can exist without assuming additional set-theoretic assumptions beyond those of ZFC) was not resolved until recently. Todorcevic proved that under PFA there a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindelöf Space
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of '' compactness'', which requires the existence of a ''finite'' subcover. A hereditarily Lindelöf space is a topological space such that every subspace of it is Lindelöf. Such a space is sometimes called strongly Lindelöf, but confusingly that terminology is sometimes used with an altogether different meaning. The term ''hereditarily Lindelöf'' is more common and unambiguous. Lindelöf spaces are named after the Finnish mathematician Ernst Leonard Lindelöf. Properties of Lindelöf spaces * Every compact space, and more generally every σ-compact space, is Lindelöf. In particular, every countable space is Lindelöf. * A Lindelöf space is compact if and only if it is countably compact. * Every second-countable space is Lindelöf, but not conversely. For example, there are many compact spaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zermelo–Fraenkel Set Theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stevo Todorcevic
Stevo ( sr, Стево) is a masculine given name and nickname. It may refer to: * Stevo Glogovac (born 1973), footballer * Stevo Karapandža (born 1947), celebrity chef * Stevo Pearce (born 1962), owner of record label Some Bizzare Records * Stevo Pendarovski (born 1963), politician * Stevo Stepanovski (born 1950), bibliophile * Stevo Teodosievski (1924–1997), musician * Stevo Todorčević (born 1955), mathematician at the University of Toronto * Stevo Žigon (1926–2005), actor * Steve Borgovini, nicknamed Steve-O, former member of the band Fun Lovin' Criminals * Steve Jocz (born 1981), nicknamed Stevo, member of the band Sum 41 * Steven Ronald Jensen, nicknamed Stevo, member of the band The Vandals * Stevica Ristić (born 1983), nicknamed Stevo, Macedonian football (soccer) player currently playing in Korea * Mike Stephenson (born 1947), nicknamed Stevo, former professional rugby league footballer, now a commentator * Stevo, a character in the film ''SLC Punk!'' See ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Forcing Axiom
In the mathematical field of set theory, the proper forcing axiom (''PFA'') is a significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings. Statement A forcing or partially ordered set P is proper if for all regular uncountable cardinals \lambda , forcing with P preserves stationary subsets of lambda\omega . The proper forcing axiom asserts that if P is proper and Dα is a dense subset of P for each α<ω1, then there is a filter G P such that Dα ∩ G is nonempty for all α<ω1. The class of proper forcings, to which PFA can be applied, is rather large. For example, standard arguments show that if P is or [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin's Axiom
In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consistent with ZFC and the negation of the continuum hypothesis. Informally, it says that all cardinals less than the cardinality of the continuum, \mathfrak c, behave roughly like \aleph_0. The intuition behind this can be understood by studying the proof of the Rasiowa–Sikorski lemma. It is a principle that is used to control certain forcing arguments. Statement For any cardinal 𝛋, we define a statement, denoted by MA(𝛋): For any partial order ''P'' satisfying the countable chain condition (hereafter ccc) and any family ''D'' of dense sets in ''P'' such that '', D, '' ≤ 𝛋, there is a filter ''F'' on ''P'' such that ''F'' ∩ ''d'' is non-empty for every ''d'' in ''D''. \operatorname(\aleph_0) is simply true — this is known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohen Real
Cohen may refer to: Places * Cohen-kuhi Tau/4, a star 420 light-years away from Earth in the Taurus Constellation *The Cohen Building of ''The Judd School'' in Tonbridge, England People * Cohen (surname), a common Jewish surname Arts, entertainment, and media *Matt Cohen Prize, an award given annually by the Writers' Trust of Canada to a Canadian writer * Shaughnessy Cohen Award, a Canadian literary award Law * Clinger–Cohen Act, a United States federal law that is designed to improve the way the federal government acquires and manages information technology *'' Cohen v. California'', a U.S. Supreme Court case dealing with freedom of speech *'' Cohen v. Cowles Media Co.'', a U.S. Supreme Court case establishing that freedom of the press does not exempt newspapers from generally applicable laws *''Cohens v. Virginia'', a U.S. Supreme Court decision most noted for the Marshall Court's assertion of its power to review state supreme court decisions in criminal law matters *'' Fla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Justin T
Justin may refer to: People * Justin (name), including a list of persons with the given name Justin * Justin (historian), a Latin historian who lived under the Roman Empire * Justin I (c. 450–527), or ''Flavius Iustinius Augustus'', Eastern Roman Emperor who ruled from 518 to 527 * Justin II (c. 520–578), or ''Flavius Iustinius Iunior Augustus'', Eastern Roman emperor who ruled from 565 to 578 * Justin (magister militum per Illyricum) (''fl.'' 538–552), a Byzantine general * Justin (Moesia), a Byzantine general killed in battle in 528 * Justin (consul 540) (c. 525–566), a Byzantine general * Justin Martyr (103–165), a Christian martyr * Justin (gnostic), 2nd-century Gnostic Christian; sometimes confused with Justin Martyr * Justin the Confessor (d 269) * Justin of Chieti, venerated as an early bishop of Chieti, Italy * Justin of Siponto (c. 4th century), venerated as Christian martyrs by the Catholic Church * Justin de Jacobis (1800–1860), an Italian Lazarist mission ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rho Functions
Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the same glyph, Ρ, as the distinct Latin letter P; the two letters have different Unicode encodings. Uses Greek Rho is classed as a liquid consonant (together with Lambda and sometimes the nasals Mu and Nu), which has important implications for morphology. In both Ancient and Modern Greek, it represents a alveolar trill , alveolar tap , or alveolar approximant . In polytonic orthography, a rho at the beginning of a word is written with a rough breathing, equivalent to ''h'' ( ''rh''), and a double rho within a word is written with a smooth breathing over the first rho and a rough breathing over the second ( ''rrh''). That apparently reflected an aspirated or voiceless pronunciation in Ancient Greek, which led to the various Greek-der ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology. The fundamental concepts in point-set topology are ''continuity'', ''compactness'', and ''connectedness'': * Continuous functions, intuitively, take nearby points to nearby points. * Compact sets are those that can be covered by finitely many sets of arbitrarily small size. * Connected sets are sets that cannot be divided into two pieces that are far apart. The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a ''t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
