|
Final Topology
In general topology and related areas of mathematics, the final topology (or coinduced, weak, colimit, or inductive topology) on a Set (mathematics), set X, with respect to a family of functions from Topological space, topological spaces into X, is the finest topology on X that makes all those functions Continuous function (topology), continuous. The Quotient space (topology), quotient topology on a quotient space is a final topology, with respect to a single surjective function, namely the quotient map. The Disjoint union (topology), disjoint union topology is the final topology with respect to the inclusion maps. The final topology is also the topology that every direct limit in the category of topological spaces is endowed with, and it is in the context of direct limits that the final topology often appears. A topology is Coherent topology, coherent with some collection of Subspace topology, subspaces if and only if it is the final topology induced by the natural inclusions. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Topology
In mathematics, general topology (or point set 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. 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 ''topology''. A set with a topology is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comparison Of Topologies
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the collection of subsets which are considered to be "open". (An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa. In the following, it doesn't matter which definition is used.) For definiteness the reader should think of a topology as the family of open sets of a topological space, since that is the standard meaning of the word "topology". Let ''τ''1 and ''τ''2 be two topologies on a set ''X'' such that ''τ''1 is contained in ''τ''2: :\tau_1 \subseteq \tau_2. That is, every element of ''τ''1 is also an element of ''τ''2. Then the topology ''τ''1 is said to b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection (set Theory)
In set theory, the intersection of two Set (mathematics), sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A. Notation and terminology Intersection is written using the symbol "\cap" between the terms; that is, in infix notation. For example: \\cap\=\ \\cap\=\varnothing \Z\cap\N=\N \\cap\N=\ The intersection of more than two sets (generalized intersection) can be written as: \bigcap_^n A_i which is similar to capital-sigma notation. For an explanation of the symbols used in this article, refer to the table of mathematical symbols. Definition The intersection of two sets A and B, denoted by A \cap B, is the set of all objects that are members of both the sets A and B. In symbols: A \cap B = \. That is, x is an element of the intersection A \cap B if and only if x is both an element of A and an element of B. For example: * The intersection of the sets and is . * The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Of Topologies
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the collection of subsets which are considered to be "open". (An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa. In the following, it doesn't matter which definition is used.) For definiteness the reader should think of a topology as the family of open sets of a topological space, since that is the standard meaning of the word "topology". Let ''τ''1 and ''τ''2 be two topologies on a set ''X'' such that ''τ''1 is contained in ''τ''2: :\tau_1 \subseteq \tau_2. That is, every element of ''τ''1 is also an element of ''τ''2. Then the topology ''τ''1 is said to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infimum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, and if ''b'' is a lower bound of S, then ''b'' is less than or equal to the infimum of S. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; : suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. If the supremum of S exists, it is unique, and if ''b'' is an upper bound of S, then the supremum of S is less than or equal to ''b''. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is, in a precise sense, dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in anal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Family Of Sets
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. Additionally, a family of sets may be defined as a function from a set I, known as the index set, to F, in which case the sets of the family are indexed by members of I. In some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a proper class. A finite family of subsets of a finite set S is also called a '' hypergraph''. The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions. Examples The set of all subsets of a given set S is called the pow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Union
In mathematics, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and appears twice in the disjoint union, with two different labels. A disjoint union of an indexed family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injective function, injection of each A_i into A, such that the image (mathematics), images of these injections form a Partition (set theory), partition of A (that is, each element of A belongs to exactly one of these images). A disjoint union of a family of pairwise disjoint sets is their Union (set theory), union. In category theory, the disjoint union is the coproduct of the category of sets, and thus defined up to a bijection. In this context, the notation \coprod_ A_i is often used. The disjoint union of two sets A and B is written with infix notation as A \sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Union (topology)
In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, in the disjoint union the given spaces are considered as part of a single new space where each looks as it would alone and they are isolated from each other. The name ''coproduct'' originates from the fact that the disjoint union is the categorical dual of the product space construction. Definition Let be a family of topological spaces indexed by ''I''. Let :X = \coprod_i X_i be the disjoint union of the underlying sets. For each ''i'' in ''I'', let :\varphi_i : X_i \to X\, be the canonical injection (defined by \varphi_i(x)=(x,i)). The disjoint union topology on ''X'' is defined as the finest topology on ''X'' for which all the canonical inj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Topology
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space. Intuitively speaking, the points of each equivalence class are or "glued together" for forming a new topological space. For example, identifying the points of a sphere that belong to the same diameter produces the projective plane as a quotient space. Definition Let X be a topological space, and let \sim be an equivalence relation on X. The quotient set Y = X/ is the set of equivalence classes of elements of X. The equivalenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
