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List Of Examples In General Topology
{{Short description, none This is a list of useful examples in general topology, a field of mathematics. * Alexandrov topology * Cantor space * Co-kappa topology ** Cocountable topology ** Cofinite topology * Compact-open topology * Compactification * Discrete topology * Double-pointed cofinite topology * Extended real number line * Finite topological space * Hawaiian earring * Hilbert cube * Irrational cable on a torus * Lakes of Wada * Long line * Order topology ** Lexicographical/dictionary order ** Ordinal number topology ** Real line ** Split interval * Overlapping interval topology * Moore plane * Sierpiński space * Sorgenfrey line * Sorgenfrey plane * Space-filling curve * Topologist's sine curve * Trivial topology * Unit interval * Zariski topology See also *Counterexamples in Topology ''Counterexamples in Topology'' (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like ... [...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 ''top ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, the order topology on ''X'' is generated by the subbase of "open rays" :\ :\ for all ''a, b'' in ''X''. Provided ''X'' has at least two elements, this is equivalent to saying that the open intervals :(a,b) = \ together with the above rays form a base for the order topology. The open sets in ''X'' are the sets that are a union of (possibly infinitely many) such open intervals and rays. A topological space ''X'' is called orderable or linearly orderable if there exists a total order on its elements such that the order topology induced by that order and the given topology on ''X'' coincide. The order topology makes ''X'' into a completely normal Hausdorff space. The standard topologies on R, Q, Z, and N are the order topologies. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: , , and . However, the notation ' is most commonly reserved for the closed interval . Properties The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space, it is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval. In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trivial Topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space is a pseudometric space in which the distance between any two points is zero. Details The trivial topology is the topology with the least possible number of open sets, namely the empty set and the entire space, since the definition of a topology requires these two sets to be open. Despite its simplicity, a space ''X'' with more than one element and the trivial topology lacks a key desirable property: it is not a T0 space. Other properties of an indiscrete space ''X''—many of which are quite unusual—include: * The only closed sets are the empty set and ''X''. * The only possible basis of ''X'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topologist's Sine Curve
In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example. It can be defined as the graph of the function sin(1/''x'') on the half-open interval (0, 1], together with the origin, under the topology subspace topology, induced from the Euclidean plane: : T = \left\ \cup \. Properties The topologist's sine curve ''T'' is connected but neither locally connected nor path connected. This is because it includes the point (0,0) but there is no way to link the function to the origin so as to make a path. The space ''T'' is the continuous image of a locally compact space (namely, let ''V'' be the space ∪ (0, 1], and use the map ''f'' from ''V'' to ''T'' defined by = (0,0) and = for ''x'' > 0), but ''T'' is not locally compact itself. The topological dimension of ''T'' is 1. Variants Two variants of the topologist's sine c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space-filling Curve
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called ''Peano curves'', but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano. Definition Intuitively, a curve in two or three (or higher) dimensions can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion, Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the precise description of the notion of a ''curve'': In the most general form, the range of such a function may lie in an arbitrary topological space, but in the most commonly studied cases, the range will lie in a Euclidean space such as the 2-dimensional plane (a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sorgenfrey Plane
In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of two copies of the Sorgenfrey line, which is the real line \mathbb under the half-open interval topology. The Sorgenfrey line and plane are named for the American mathematician Robert Sorgenfrey. A basis for the Sorgenfrey plane, denoted \mathbb from now on, is therefore the set of rectangles that include the west edge, southwest corner, and south edge, and omit the southeast corner, east edge, northeast corner, north edge, and northwest corner. Open sets in \mathbb are unions of such rectangles. \mathbb is an example of a space that is a product of Lindelöf spaces that is not itself a Lindelöf space. The so-called anti-diagonal \Delta = \ is an uncountable discrete subset of this space, and this is a non- separable subset of the separable space \mathbb. It shows that separability does not inherit to closed subspaces. Note t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lower Limit Topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of interesting properties. It is the topology generated by the basis of all half-open intervals ''a'',''b''),_where_''a''_and_''b''_are_real_numbers. The_resulting_topological_space.html" ;"title="/nowiki>''a'',''b''), where ''a'' and ''b'' are real numbers. The resulting topological space">/nowiki>''a'',''b''), where ''a'' and ''b'' are real numbers. The resulting topological space is called the Sorgenfrey line after Robert Sorgenfrey or the arrow and is sometimes written \mathbb_l. Like the Cantor set and the long line (topology), long line, the Sorgenfrey line often serves as a useful counterexample to many otherwise plausible-sounding conjectures in general topology. The product of \mathbb_l with itself is also a useful counterexample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sierpiński Space
In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński. The Sierpiński space has important relations to the theory of computation and semantics, because it is the classifying space for open sets in the Scott topology. Definition and fundamental properties Explicitly, the Sierpiński space is a topological space ''S'' whose underlying point set is \ and whose open sets are \. The closed sets are \. So the singleton set \ is closed and the set \ is open (\varnothing = \ is the empty set). The closure operator on ''S'' is determined by \overline = \, \qquad \overline = \. A finite topological space is also uniquely determined by its specialization preorder. For the Sierpiński space this preorder is actually a partial order and given by 0 \leq 0, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moore Plane
In mathematics, the Moore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii's tangent disk topology), is a topological space. It is a completely regular Hausdorff space (also called Tychonoff space) that is not normal. It is named after Robert Lee Moore and Viktor Vladimirovich Nemytskii. Definition If \Gamma is the (closed) upper half-plane \Gamma = \, then a topology may be defined on \Gamma by taking a local basis \mathcal(p,q) as follows: *Elements of the local basis at points (x,y) with y>0 are the open discs in the plane which are small enough to lie within \Gamma. *Elements of the local basis at points p = (x,0) are sets \\cup A where ''A'' is an open disc in the upper half-plane which is tangent to the ''x'' axis at ''p''. That is, the local basis is given by :\mathcal(p,q) = \begin \, & \mbox q > 0; \\ \, & \mbox q = 0. \end Thus the subspace topology inherited by \Gamma\backslash \ is the same as the subspace topology inherited from t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Overlapping Interval Topology
In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles. Definition Given the closed interval 1,1/math> of the real number line, the open sets of the topology are generated from the half-open intervals (a,1] with a 0. The topology therefore consists of intervals of the form 1,b)_with_b_>_0._The_topology_therefore_consists_of_intervals_of_the_form_[-1,b),_(a,b),_and_(a,1/math>_with_a_1,b) with b > 0. The topology therefore consists of intervals of the form [-1,b), (a,b), and (a,1/math> with a distinct points in 1,1/math> are topologically distinguishable under the overlapping interval topology as one can always find an open set containing one but not the other point. However, every non-empty open set contains the point 0 which can therefore not be separated from any other point in 1,1/math>, making 1,1/math> with the overlapping interval topology an example of a T0 space that is not a T1 space. The overlap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Split Interval
In topology, the split interval, or double arrow space, is a topological space that results from splitting each point in a closed interval into two adjacent points and giving the resulting ordered set the order topology. It satisfies various interesting properties and serves as a useful counterexample in general topology. Definition The split interval can be defined as the lexicographic product , 1\times\ equipped with the order topology. Equivalently, the space can be constructed by taking the closed interval ,1/math> with its usual order, splitting each point a into two adjacent points a^- [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
