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Polyadic Space
In mathematics, a polyadic space is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space. History Polyadic spaces were first studied by S. Mrówka in 1970 as a generalisation of dyadic spaces. The theory was developed further by R. H. Marty, János Gerlits and Murray G. Bell, the latter of whom introduced the concept of the more general centred spaces. Background A subset ''K'' of a topological space ''X'' is said to be compact if every open cover of ''K'' contains a finite subcover. It is said to be locally compact at a point ''x'' ∈ ''X'' if ''x'' lies in the interior of some compact subset of ''X''. ''X'' is a locally compact space if it is locally compact at every point in the space. A proper subset ''A'' ⊂ ''X'' is said to be dense if the closure ''Ā'' = ''X''. A space whose set has a countable, dense subset is called a separable space. For a non-compact, locally c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinal Number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ''transfinite'' cardinal numbers, often denoted using the Hebrew symbol \aleph ( aleph) followed by a subscript, describe the sizes of infinite sets. Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compactness Number
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Tennessee At Martin
The University of Tennessee at Martin (UT Martin or UTM) is a public university in Martin, Tennessee. It is one of the five campuses of the University of Tennessee system. UTM is the only public university in West Tennessee outside of Memphis. UTM operates a large experimental farm and several satellite centers in West Tennessee. History Although UT Martin dates from 1927, it is not the first educational institution to use the current site. In 1900, Ada Gardner Brooks donated a site on what was then the outskirts of Martin to the Tennessee Baptist Convention for the purposes of opening a school. The school opened as the Hall-Moody Institute, named for two locally prominent Baptist ministers - John Newton Hall and Joseph Burnley Moody. It originally offered 13 years of study, from elementary grades to the equivalent of the first years of collegiate work. The institute changed its name to Hall-Moody Normal School in 1917, as teacher training became its primary focus. Five ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clopen
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open set, open and closed set, closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its Complement (set theory), complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open closed, and therefore clopen. As described by topologist James Munkres, unlike a door, "a set can be open, or closed, or both, or neither!" emphasizing that the meaning of "open"/"closed" for is unrelated to their meaning for (and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as "door spaces" their name. Examples In any topological space X, the empty set and the whole space X are both clopen. Now consider the spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Space
In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Marshall H. Stone. Stone was led to it by his study of the spectral theory of operators on a Hilbert space. Stone spaces Each Boolean algebra ''B'' has an associated topological space, denoted here ''S''(''B''), called its Stone space. The points in ''S''(''B'') are the ultrafilters on ''B'', or equivalently the homomorphisms from ''B'' to the two-element Boolean algebra. The topology on ''S''(''B'') is generated by a (closed) basis consisting of all sets of the form \, where ''b'' is an element of ''B''. This is the topology of pointwise convergence of nets of homomorphisms into the two-element Boolean algebra. For every Boolean algebra ''B'', ''S''(''B'') is a compact totally disco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramsey's Theorem
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say, blue and red), let and be any two positive integers. Ramsey's theorem states that there exists a least positive integer for which every blue-red edge colouring of the complete graph on vertices contains a blue clique on vertices or a red clique on vertices. (Here signifies an integer that depends on both and .) Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by F. P. Ramsey. This initiated the combinatorial theory now called Ramsey theory, that seeks regularity amid disorder: general conditions for the existence of substructures with regular properties. In this application it is a question of the existence of ''monochromatic subsets'', that is, subsets of connected edges of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ohio University
Ohio University is a Public university, public research university in Athens, Ohio. The first university chartered by an Act of Congress and the first to be chartered in Ohio, the university was chartered in 1787 by the Congress of the Confederation and subsequently approved for the territory in 1802 and state in 1804, opening for students in 1809. Ohio University is the oldest university in Ohio and among the oldest public universities in the United States. Ohio University comprises nine campuses, nine undergraduate colleges, its Graduate College, its college of medicine, and its public affairs school, and offers more than 250 areas of undergraduate study as well as certificates, master's, and doctoral degrees. The university is accredited by the Higher Learning Commission and Carnegie Classification of Institutions of Higher Education, classified among List of research universities in the United States#Universities classified as "R1: Doctoral Universities – Very high resear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suslin Property
In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent. (Suslin is also sometimes written with the French transliteration as , from the Cyrillic .) Formulation Suslin's problem asks: Given a non-empty totally ordered set ''R'' with the four properties # ''R'' does not have a least nor a greatest element; # the order on ''R'' is dense (between any two distinct elements there is another); # the order on ''R'' is complete, in the sense that every non-empty bounded subset has a supremum and an infimum; and # every collection of mutually disjoint non-empty open intervals in ''R'' is countable (this is the countable chain condition for the order topology of ''R''), is ''R'' necessarily order-isomorphic to the real line R? ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base (topology)
In mathematics, a base (or basis) for the topology of a topological space is a family \mathcal of open subsets of such that every open set of the topology is equal to the union of some sub-family of \mathcal. For example, the set of all open intervals in the real number line \R is a basis for the Euclidean topology on \R because every open interval is an open set, and also every open subset of \R can be written as a union of some family of open intervals. Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called , are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier. Not all families of subsets of a set X form a base for a topology on X. Under some cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Auburn University
Auburn University (AU or Auburn) is a public land-grant research university in Auburn, Alabama. With more than 24,600 undergraduate students and a total enrollment of more than 30,000 with 1,330 faculty members, Auburn is the second largest university in Alabama. It is one of the state's two public flagship universities. The university is classified among "R1: Doctoral Universities – Very High Research Activity" and its alumni include 5 Rhodes Scholars and 5 Truman Scholars. Auburn was chartered on February 1, 1856, as East Alabama Male College, a private liberal arts school affiliated with the Methodist Episcopal Church, South. In 1872, under the Morrill Act, it became the state's first land-grant university and was renamed as the Agricultural and Mechanical College of Alabama. In 1892, it became the first four-year coeducational school in Alabama, and in 1899 was renamed Alabama Polytechnic Institute (API) to reflect its changing mission. In 1960, its name was changed t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of East Anglia
The University of East Anglia (UEA) is a public research university in Norwich, England. Established in 1963 on a campus west of the city centre, the university has four faculties and 26 schools of study. The annual income of the institution for 2020–21 was £292.1 million, of which £35.2 million was from research grants and contracts, with an expenditure of £290.4 million, and had an undergraduate offer rate of 85.1% in 2021. UEA alumni and faculty include three Nobel laureates, a discoverer of Hepatitis C and of the Hepatitis D genome, a lead developer of the Oxford–AstraZeneca COVID-19 vaccine, one President of the Royal Society, and at least 48 Fellows of the Royal Society. Alumni also include heads of state, government and intergovernmental organisations, as well as three Booker Prize winning authors. History 1960s People in Norwich began to talk about the possibility of setting up a university in the nineteenth century, and attempts to establish ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
