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CLRg Property
In mathematics, the notion of ''“common limit in the range”'' property denoted by CLRg property is a theorem that unifies, generalizes, and extends the contractive mappings in fuzzy metric spaces, where the range of the mappings does not necessarily need to be a closed subspace of a non-empty set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ... X. Suppose X is a non-empty set, and d is a distance metric; thus, (X, d) is a metric space. Now suppose we have self mappings f,g : X \to X. These mappings are said to fulfil CLRg property if \lim_ f x_ = \lim_ g x_ = gx, for some x \in X. Next, we give some examples that satisfy the CLRg property. Examples Source: Example 1 Suppose (X,d) is a usual metric space, with X=[0,\infty). Now, if the mappings f,g: X \to X are de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contractive Mapping
In mathematics, a contraction mapping, or contraction or contractor, on a metric space (''M'', ''d'') is a function ''f'' from ''M'' to itself, with the property that there is some real number 0 \leq k < 1 such that for all ''x'' and ''y'' in ''M'', : The smallest such value of ''k'' is called the Lipschitz constant of ''f''. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied for ''k'' ≤ 1, then the mapping is said to be a non-expansive map. More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (''M'', ''d'') and (''N'', ''d) are two metric spaces, then is a contractive mapping if there is a constant such that : |
Fuzzy Metric Spaces
Fuzzy or Fuzzies may refer to: Music * Fuzzy (band), a 1990s Boston indie pop band * Fuzzy (composer), Danish composer Jens Vilhelm Pedersen (born 1939) * ''Fuzzy'' (album), 1993 debut album of American rock band Grant Lee Buffalo * "Fuzzy", a song from the 2009 ''Collective Soul'' album by Collective Soul * "Fuzzy", a song from '' Poppy.Computer'', the debut 2017 album by Poppy * Fuzzy, an Australian events company that organises Listen Out, a multi-city Australian music festival Nickname * Faustina Agolley (born 1984), Australian television presenter, host of the Australian television show ''Video Hits'' * Fuzzy Haskins (1941–2023), American singer and guitarist with the doo-wop group Parliament-Funkadelic * Fuzzy Hufft (1901−1973), American baseball player * Fuzzy Knight (1901−1976), American actor * Andrew Levane (1920−2012), American National Basketball Association player and coach * Robert Alfred Theobald (1884−1957), United States Navy rear admiral * Fuzzy Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Map (mathematics)
In mathematics, a map or mapping is a function (mathematics), function in its general sense. These terms may have originated as from the process of making a map, geographical map: ''mapping'' the Earth surface to a sheet of paper. The term ''map'' may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial. In category theory, a map may refer to a morphism. The term ''transformation'' can be used interchangeably, but ''transformation (function), transformation'' often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory. Maps as functions In many branches of mathematics, the term ''map'' is used to mean a Function (mathematics), function, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "continuous f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is Closure (mathematics), closed under the limit of a sequence, limit operation. This should not be confused with closed manifold. Sets that are both open and closed and are called clopen sets. Definition Given a topological space (X, \tau), the following statements are equivalent: # a set A \subseteq X is in X. # A^c = X \setminus A is an open subset of (X, \tau); that is, A^ \in \tau. # A is equal to its Closure (topology), closure in X. # A contains all of its limit points. # A contains all of its Boundary (topology), boundary points. An alternative characterization (mathematics), characterization of closed sets is available via sequences and Net (mathematics), net ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Subspace (topology)
In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology, or the induced topology, or the trace topology).; see Section 26.2.4. Submanifolds, p. 59 Definition Given a topological space (X, \tau) and a subset S of X, the subspace topology on S is defined by :\tau_S = \lbrace S \cap U \mid U \in \tau \rbrace. That is, a subset of S is open in the subspace topology if and only if it is the intersection of S with an open set in (X, \tau). If S is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of (X, \tau). Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated. Alternatively we can define the subspace topology for a subset S of X as the coarsest topology for which the inclusi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
In mathematics, a set is a collection of different things; the things are '' elements'' or ''members'' of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, or other 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 are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Context Before the end of the 19th century, sets were not studied specifically, and were not clearly distinguished from sequences. Most mathematicians considered infinity as potentialmeaning that it is the result of an endless processand were reluctant to consider infinite sets, that is sets whose number of members is not a natural number. Specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
