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Baire Space (set Theory)
In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology, called the product topology. This space is commonly used in descriptive set theory, to the extent that its elements are often called "reals". It is denoted by \N^, or ωω, or by the symbol \mathcal or sometimes by ωω (not to be confused with the countable ordinal obtained by ordinal exponentiation). The Baire space is defined to be the Cartesian product of countably infinitely many copies of the set of natural numbers, and is given the product topology (where each copy of the set of natural numbers is given the discrete topology). The Baire space is often represented using the tree of finite sequences of natural numbers. (This space should also not be confused with the concept of a Baire space, which is a certain kind of topological space.) The Baire space can be contrasted with Cantor space, the set of infinite sequences of binary digits. Topology and trees ... [...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] |
Shortlex
In mathematics, and particularly in the theory of formal languages, shortlex is a total ordering for finite sequences of objects that can themselves be totally ordered. In the shortlex ordering, sequences are primarily sorted by cardinality (length) with the shortest sequences first, and sequences of the same length are sorted into lexicographical order. Shortlex ordering is also called radix, length-lexicographic, military, or genealogical ordering. In the context of strings on a totally ordered alphabet, the shortlex order is identical to the lexicographical order, except that shorter strings precede longer strings. For example, the shortlex order of the set of strings on the English alphabet (in its usual order) is 'ε, a, b, c, ..., z, aa, ab, ac, ..., zz, aaa, aab, aac, ..., zzz, ...'' where ε denotes the empty string. The strings in this ordering over a fixed finite alphabet can be placed into one-to-one order-preserving correspondence with the natural numbers, giving the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Metric Space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. \sqrt is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the ''completion'' of a given space, as explained below. Definition Cauchy sequence A sequence x_1, x_2, x_3, \ldots of elements from X of a metric space (X, d) is called Cauchy if for every positive real number r > 0 there is a positive integer N such that for all positive integers m, n > N, d(x_m, x_n) < r. Complete space A metric space is complete if any of the following equivalent conditions are satisfied: #Every Cauchy seq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polish Space
In the mathematical discipline of general topology, a Polish space is a separable space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that has a countable Dense set, dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Alfred Tarski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory. Common examples of Polish spaces are the real line, any Separable space, separable Banach space, the Cantor space, and the Baire space (set theory), Baire space. Additionally, some spaces that are not complete metric spaces in the usual metric may be Polish; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Set
In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set S is perfect if S=S', where S' denotes the set of all limit points of S, also known as the derived set of S. (Some authors do not consider the empty set to be perfect.) In a perfect set, every point can be approximated arbitrarily well by other points from the set: given any point of S and any neighborhood of the point, there is another point of S that lies within the neighborhood. Furthermore, any point of the space that can be so approximated by points of S belongs to S. Note that the term ''perfect space'' is also used, incompatibly, to refer to other properties of a topological space, such as being a Gδ space. As another possible source of confusion, also note that having the perfect set property is not the same as being a perfect set. Examples Examples of perfect subsets of the real line \mathbb are the empty set, all closed intervals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is given by some Injective function, injective and structure-preserving map f:X\rightarrow Y. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances. In the terminology of category theory, a structure-preserving map is called a morphism. The fact that a map f:X\rightarrow Y is an embedding is often indicated by the use of a "hooked arrow" (); thus: f : X \hookrightarrow Y. (On the other hand, this notation is sometimes reserved for inclusion maps.) Given X and Y, several different embeddings of X in Y may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ball (mathematics)
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ''ball'' in dimensions is called a hyperball or -ball and is bounded by a ''hypersphere'' or ()-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the planar region bounded by a circle. In Euclidean 3-space, a ball is taken to be the region of space bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, ''sphere'' is sometimes used to mean ''ball''. In the field of topology the closed n-dimensional ball is often denoted as B^n or D^n while the open n-dimensional ball is \o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a Conceptual metaphor , metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thumb is ''pollex'' (compare ''hallux'' for big toe), and the corresponding adjective for thumb is ''pollical''. Definition Thumb and fingers The English word ''finger'' has two senses, even in the context of appendages of a single typical human hand: 1) Any of the five terminal members of the hand. 2) Any of the four terminal members of the hand, other than the thumb. Linguistically, it appears that the original sense was the first of these two: (also rendered as ) was, in the inferred Proto-Indo-European language, a suffixed form of (or ), which has given rise to many Indo-European-family words (tens of them defined in English dictionaries) that involve, or stem from, concepts of fiveness. The thumb shares the following with each of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Box Topology
In topology, the cartesian product of topological spaces can be given several different topologies. One of the more natural choices is the box topology, where a base is given by the Cartesian products of open sets in the component spaces. Another possibility is the product topology, where a base is also given by the Cartesian products of open sets in the component spaces, but only finitely many of which can be unequal to the entire component space. While the box topology has a somewhat more intuitive definition than the product topology, it satisfies fewer desirable properties. In particular, if all the component spaces are compact, the box topology on their Cartesian product will not necessarily be compact, although the product topology on their Cartesian product will always be compact. In general, the box topology is finer than the product topology, although the two agree in the case of finite direct products (or when all but finitely many of the factors are trivial). Definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Hierarchy
In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called the rank of the Borel set. The Borel hierarchy is of particular interest in descriptive set theory. One common use of the Borel hierarchy is to prove facts about the Borel sets using transfinite induction on rank. Properties of sets of small finite ranks are important in measure theory and analysis. Borel sets The Borel algebra in an arbitrary topological space is the smallest collection of subsets of the space that contains the open sets and is closed under countable unions and complementation. It can be shown that the Borel algebra is closed under countable intersections as well. A short proof that the Borel algebra is well-defined proceeds by showing that the entire powerset of the space is closed under complements and co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointclass
In the mathematical field of descriptive set theory, a pointclass is a collection of Set (mathematics), sets of point (mathematics), points, where a ''point'' is ordinarily understood to be an element of some perfect set, perfect Polish space. In practice, a pointclass is usually characterized by some sort of ''definability property''; for example, the collection of all open sets in some fixed collection of Polish spaces is a pointclass. (An open set may be seen as in some sense definable because it cannot be a purely arbitrary collection of points; for any point in the set, all points sufficiently close to that point must also be in the set.) Pointclasses find application in formulating many important principles and theorems from set theory and real analysis. Strong set-theoretic principles may be stated in terms of the determinacy of various pointclasses, which in turn implies that sets in those pointclasses (or sometimes larger ones) have regularity properties such as Lebesgue m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
