|
Divisor Topology
In mathematics, more specifically general topology, the divisor topology is a specific topology on the set X = \ of positive integers greater than or equal to two. The divisor topology is the poset topology for the partial order relation of divisibility of integers on X. Construction The sets S_n = \ for n = 2,3,... form a basis for the divisor topologySteen & Seebach, example 57, p. 79-80 on X, where the notation x\mathop, n means x is a divisor of n. The open sets in this topology are the lower sets for the partial order defined by x\leq y if x\mathop, y. The closed sets are the upper sets for this partial order. Properties All the properties below are proved in or follow directly from the definitions. * The closure of a point x\in X is the set of all multiples of x. * Given a point x\in X, there is a smallest neighborhood of x, namely the basic open set S_x of divisors of x. So the divisor topology is an Alexandrov topology. * X is a T0 space. Indeed, given two points ... [...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] |
Second-countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mathcal = \_^ of open subsets of T such that any open subset of T can be written as a union of elements of some subfamily of \mathcal. A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have. Many "well-behaved" spaces in mathematics are second-countable. For example, Euclidean space (R''n'') with its usual topology is second-countable. Although the usual base of open balls is uncountable, one can restrict to the collection of all open balls with rational radii and whose centers have rational coordinates. This restricted set is countable and still forms a basis. Properties Second-countability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Compact
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. In mathematical analysis locally compact spaces that are Hausdorff are of particular interest; they are abbreviated as LCH spaces. Formal definition Let ''X'' be a topological space. Most commonly ''X'' is called locally compact if every point ''x'' of ''X'' has a compact neighbourhood, i.e., there exists an open set ''U'' and a compact set ''K'', such that x\in U\subseteq K. There are other common definitions: They are all equivalent if ''X'' is a Hausdorff space (or preregular). But they are not equivalent in general: :1. every point of ''X'' has a compact neighbourhood. :2. every point of ''X'' has a closed compact neighbourhood. :2′. every point of ''X'' has a relatively compact neighbourhoo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
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 topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Subset
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 topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scattered Space
In mathematics, a scattered space is a topological space ''X'' that contains no nonempty dense-in-itself subset. Equivalently, every nonempty subset ''A'' of ''X'' contains a point isolated in ''A''. A subset of a topological space is called a scattered set if it is a scattered space with the subspace topology. Examples * Every discrete space is scattered. * Every ordinal number with the order topology is scattered. Indeed, every nonempty subset ''A'' contains a minimum element, and that element is isolated in ''A''. * A space ''X'' with the particular point topology, in particular the Sierpinski space, is scattered. This is an example of a scattered space that is not a T1 space. * The closure of a scattered set is not necessarily scattered. For example, in the Euclidean plane \R^2 take a countably infinite discrete set ''A'' in the unit disk, with the points getting denser and denser as one approaches the boundary. For example, take the union of the vertices of a serie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Path Connected
In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness and compactness have been two of the most widely studied topological properties. Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space. However, whereas the structure of ''compact'' subsets of Euclidean space was understood quite early on via the Heine–Borel theorem, ''connected'' subsets of \R^n (for ''n'' > 1) proved to be much more complicated. Indeed, while any compact Hausdorff space is locally compact, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is ''locally connected'', which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. For a topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Connected
In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness and compactness have been two of the most widely studied topological properties. Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space. However, whereas the structure of ''compact'' subsets of Euclidean space was understood quite early on via the Heine–Borel theorem, ''connected'' subsets of \R^n (for ''n'' > 1) proved to be much more complicated. Indeed, while any compact Hausdorff space is locally compact, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is ''locally connected'', which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. For a topologi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Space
In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' can be separated by neighborhoods. This condition is known as Axiom T3. The term "T3 space" usually means "a regular Hausdorff space". These conditions are examples of separation axioms. Definitions A topological space ''X'' is a regular space if, given any closed set ''F'' and any point ''x'' that does not belong to ''F'', there exists a neighbourhood ''U'' of ''x'' and a neighbourhood ''V'' of ''F'' that are disjoint. Concisely put, it must be possible to separate ''x'' and ''F'' with disjoint neighborhoods. A or is a topological space that is both regular and a Hausdorff space. (A Hausdorff space or T2 space is a topological space in which any two distinct points are separated by neighbourhoods.) It turns out that a space is T3 if a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separated Set
In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of connected spaces (and their connected components) as well as to the separation axioms for topological spaces. Separated sets should not be confused with separated spaces (defined below), which are somewhat related but different. Separable spaces are again a completely different topological concept. Definitions There are various ways in which two subsets of a topological space ''X'' can be considered to be separated. * ''A'' and ''B'' are disjoint if their intersection is the empty set. This property has nothing to do with topology as such, but only set theory. It is included here because it is the weakest in the sequence of different notions. ** ''A'' and ''B'' are separated in ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
_sine_curve.png "Click for more on -> Path Connected")