Point-set topology

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Compact support

A function has compact support if it is zero outside of a compact set. Alternatively, one can say that a function has compact support if its support is a compact set. For example, the function in its entire domain (i.e., ) does not have compact support, while any bump function does have compact support.

Semilocally simply connected

A topological space is semilocally simply connected (also called semilocally 1-connected) if every point has a neighborhood such that any loop with basepoint is homotopic to the trivial loop. The prefix semi- refers to the fact that the homotopy which takes to the trivial loop can leave and travel to other parts of .The property of semilocal simple connectedness is important because it is a necessary and sufficient condition for a connected, locally pathwise-connected space to have a universal cover.

Closed set

There are several equivalent definitions of a closed set. Let be a subset of a metric space. A set is closed if 1. The complement of is an open set, 2. is its own set closure, 3. Sequences/nets/filters in that converge do so within , 4. Every point outside has a neighborhood disjoint from . The point-set topological definition of a closed set is a set which contains all of its limit points. Therefore, a closed set is one for which, whatever point is picked outside of , can always be isolated in some open set which doesn't touch .The most commonly encountered closed sets are the closed interval, closed path, closed disk, interior of a closed path together with the path itself, and closed ball. The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points (and is nowhere dense, so it has Lebesgue measure 0).It is possible for a set to be neither open nor closed, e.g., the half-closed interval ...

Closed disk

An -dimensional closed disk of radius is the collection of points of distance from a fixed point in -dimensional Euclidean space. Krantz (1999, p. 3) uses the symbol to denote the closed disk, and to denote the unit closed disk centered at the origin

Relative topology

The topology induced by a topological space on a subset . The open sets of are the intersections , where is an open set of .For example, in the relative topology of the interval induced by the Euclidean topology of the real line, the half-open interval is open since it coincides with . This example shows that an open set of the relative topology of need not be open in the topology of .

Closed

A mathematical structure is said to be closed under an operation if, whenever and are both elements of , then so is .A mathematical object taken together with its boundary is also called closed. For example, while the interior of a sphere is an open ball, the interior together with the sphere itself is a closed ball.

Reinhardt domain

A Reinhardt domain with center is a domain in such that whenever contains , the domain also contains the closed polydisk.

Regular borel measure

An outer measure on is Borel regular if, for each set , there exists a Borel set such that . The -dimensional Hausdorff outer measure is regular on .

Generalized diameter

The generalized diameter is the greatest distance between any two points on the boundary of a closed figure. The diameter of a subset of a Euclidean space is therefore given bywhere denotes the supremum (Croft et al. 1991).For a solid object or set of points in Euclidean -space, the generalized diameter is equal to the generalized diameter of its convex hull. This means, for example, that the generalized diameter of a polygon or polyhedron can be found simply by finding the greatest distance between any two pairs of vertices (without needing to consider other boundary points).The generalized diameter is related to the geometricspan of a set of points.

Carathéodory's fundamental theorem

Each point in the convex hull of a set in is in the convex combination of or fewer points of .

Cantor's discontinuum

A Cartesian product of any finite or infinite set of copies of , equipped with the product topology derived from the discrete topology of . It is denoted . The name is due to the fact that for , this set is closely related to the Cantor set (which is formed by all numbers of the interval which admit an expansion in base 3 formed by 0s and 2s only), and this gives rise to a one-to-one correspondence between and the Cantor set, which is actually a homeomorphism. In the symbol denoting the Cantor discontinuum, can be replaced by 2 and by .

Foliation leaf

Let be an -manifold and let denote a partition of into disjoint pathwise-connected subsets. Then if is a foliation of , each is called a leaf and is not necessarily closed or compact.

Bounded set

A set in a metric space is bounded if it has a finite generalized diameter, i.e., there is an such that for all . A set in is bounded iff it is contained inside some ball of finite radius (Adams 1994).

Foliation

Let be an -manifold and let denote a partition of into disjoint pathwise-connected subsets. Then is called a foliation of of codimension (with ) if there exists a cover of by open sets , each equipped with a homeomorphism or which throws each nonempty component of onto a parallel translation of the standard hyperplane in . Each is then called a foliation leaf and is not necessarily closed or compact (Rolfsen 1976, p. 284).

Disconnected space

A topological space that is not connected, i.e., which can be decomposed as the disjoint union of two nonempty open subsets. Equivalently, it can be characterized as a space with more than one connected component.A subset of the Euclidean plane with more than one element can always be disconnected by cutting it through with a line (i.e., by taking out its intersection with a suitable straight line). In fact, it is certainly possible to find a line such that two points of lie on different sides of . If the Cartesian equation of is(1)for fixed real numbers , then the set is disconnected, since it is the union of the two nonempty open subsets(2)and(3)which are the sets of elements of lying on the two sides of .

Bicollared

A subset is said to be bicollared in if there exists an embedding such that when . The map or its image is then said to be the bicollar.

Dimension axiom

One of the Eilenberg-Steenrod axioms. Let be a single point space. unless , in which case where are some groups. The are called the coefficients of the homology .

Order topology

A topology defined on a totally ordered set whose open sets are all the finite intersections of subsets of the form or , where .The order topology of the real line is the Euclidean topology. The order topology of is the discrete topology, since for all ,is an open set.

Open set

Let be a subset of a metric space. Then the set is open if every point in has a neighborhood lying in the set. An open set of radius and center is the set of all points such that , and is denoted . In one-space, the open set is an open interval. In two-space, the open set is a disk. In three-space, the open set is a ball.More generally, given a topology (consisting of a set and a collection of subsets ), a set is said to be open if it is in . Therefore, while it is not possible for a set to be both finite and open in the topology of the real line (a single point is a closed set), it is possible for a more general topological set to be both finite and open.The complement of an open set is a closed set. It is possible for a set to be neither open nor closed, e.g., the half-closed interval ...

Dense

A set in a first-countable space is dense in if , where is the set of limit points of . For example, the rational numbers are dense in the reals. In general, a subset of is dense if its set closure .A real number is said to be -dense iff, in the base- expansion of , every possible finite string of consecutive digits appears. If is -normal, then is also -dense. If, for some , is -dense, then is irrational. Finally, is -dense iff the sequence is dense (Bailey and Crandall 2001, 2003).

Arc component

Let be a topological space, and let . Then the arc component of is

Covering space

Suppose that are arcwise-connected and locally arcwise-connected topological spaces. Then is said to be a covering space of if is a surjective continuous map with every having an open neighborhood such that every connected component of is mapped homeomorphically onto by .

Almost everywhere convergence

A weakened version of pointwise convergence hypothesis which states that, for a measure space, for all , where is a measurable subset of such that .

Hausdorff axioms

The axioms formulated by Hausdorff (1919) for his concept of a topological space. These axioms describe the properties satisfied by subsets of elements in a neighborhood set of . 1. There corresponds to each point at least one neighborhood , and each neighborhood contains the point . 2. If and are two neighborhoods of the same point , there must exist a neighborhood that is a subset of both. 3. If the point lies in , there must exist a neighborhood that is a subset of . 4. For two different points and , there are two corresponding neighborhoods and with no points in common.

Lebesgue covering dimension

The Lebesgue covering dimension is an important dimension and one of the first dimensions investigated. It is defined in terms of covering sets, and is therefore also called the covering dimension (as well as the topological dimension).A space has Lebesgue covering dimension if for every open cover of that space, there is an open cover that refines it such that the refinement has order at most . Consider how many elements of the cover contain a given point in a base space. If this has a maximum over all the points in the base space, then this maximum is called the order of the cover. If a space does not have Lebesgue covering dimension for any , it is said to be infinite dimensional.Results of this definition are: 1. Two homeomorphic spaces have the same dimension, 2. has dimension , 3. A topological space can be embedded as a closed subspace of a Euclidean space iff it is locally compact, T2, second countable, and is finite-dimensional (in the sense of the..

Bounded from below

A set is said to be bounded from below if it has a lowerbound.Consider the real numbers with their usual order. Then for any set , the infimum exists (in ) if and only if is bounded from below and nonempty.

Uhl

Ramification index

For a point , with , the ramification index of at is a positive integer such that there is some open neighborhood of so that has only one preimage in , i.e., , and for all other points , . In other words, the map from to is to 1 except at . At all but finitely many points of , we have . Note that for any point we have . Sometimes the ramification index of at is called the valency of .

Bounded from above

A set is said to be bounded from above if it has an upperbound.Consider the real numbers with their usual order. Then for any set , the supremum exists (in ) if and only if is bounded from above and nonempty.

Uhl

Fiber

A fiber of a map is the preimage of an element . That is,For instance, let and be the complex numbers . When , every fiber consists of two points , except for the fiber over 0, which has one point. Note that a fiber may be the empty set.In special cases, the fiber may be independent, in some sense, of the choice of . For instance, if is a covering map, then the fibers are all discrete and have the same cardinal number. The example is a covering map away from zero, i.e., from the punctured plane to itself has a fiber consisting of two points.When is a fiber bundle, then every fiber is isomorphic, in whatever category is being used. For instance, when is a real vector bundle of bundle rank , every fiber is isomorphic to .

Boundary set

A (symmetrical) boundary set of radius and center is the set of all points such thatLet be the origin. In , the boundary set is then the pair of points and . In , the boundary set is a circle. In , the boundary set is a sphere.

Boundary point

A point which is a member of the set closure of a given set and the set closure of its complement set. If is a subset of , then a point is a boundary point of if every neighborhood of contains at least one point in and at least one point not in .

Discrete topology

A topology is given by a collection of subsets of a topological space . The smallest topology has two open sets, the empty set and . The largest topology contains all subsets as open sets, and is called the discrete topology. In particular, every point in is an open set in the discrete topology.

Borel set

A Borel set is an element of a Borel sigma-algebra. Roughly speaking, Borel sets are the sets that can be constructed from open or closed sets by repeatedly taking countable unions and intersections. Formally, the class of Borel sets in Euclidean is the smallest collection of sets that includes the open and closed sets such that if , , , ... are in , then so are , , and , where is a set difference (Croft et al. 1991).The set of rational numbers is a Borel set, as is the Cantorset.

Discrete set

A set is discrete in a larger topological space if every point has a neighborhood such that . The points of are then said to be isolated (Krantz 1999, p. 63). Typically, a discrete set is either finite or countably infinite. For example, the set of integers is discrete on the real line. Another example of an infinite discrete set is the set . On any reasonable space, a finite set is discrete. A set is discrete if it has the discrete topology, that is, if every subset is open.In the case of a subset , as in the examples above, one uses the relative topology on . Sometimes a discrete set is also closed. Then there cannot be any accumulation points of a discrete set. On a compact set such as the sphere, a closed discrete set must be finite because of this.

Domain

The term domain has (at least) three different meanings in mathematics.The term domain is most commonly used to describe the set of values for which a function (map, transformation, etc.) is defined. For example, a function that is defined for real values has domain , and is sometimes said to be "a function over the reals." The set of values to which is sent by the function is then called the range.Unfortunately, the term range is sometimes used in probability theory to mean domain (Feller 1968, p. 200; Evans et al. 2000). To confuse matters even more, the term "range" is more commonly used in statistics to refer to a completely different quantity, known in this work as the statistical range. As if this wasn't confusing enough, Evans et al. (2000, p. 6) define a probability domain to be the range of the distribution function of a probability density function.The domain (in its usual established mathematical sense)..

Pseudoconvex function

Given a subset and a real function which is Gâteaux differentiable at a point , is said to be pseudoconvex at ifHere, denotes the usual gradient of .The term pseudoconvex is used to describe the fact that such functions share many properties of convex functions, particularly with regards to derivative properties and finding local extrema. Note, however, that pseudoconvexity is strictly weaker than convexity as every convex function is pseudoconvex though one easily checks that is pseudoconvex and non-convex.Similarly, every pseudoconvex function is quasi-convex, though the function is quasi-convex and not pseudoconvex.A function for which is pseudoconvex is said to be pseudoconcave.

Contingent cone

Given a subset and a point , the contingent cone at with respect to is defined to be the setwhere is the upper left Dini derivative of the distance functionA classical result in convex analysis characterizes as the collection of vectors in for which there are sequences in and in such that lies in for all (Borwein). Intuitively, then, the contingent cone consists of limits of directions to points near in .

Metric topology

A topology induced by the metric defined on a metric space . The open sets are all subsets that can be realized as the unions of open ballswhere , and .The metric topology makes a T2-space. Given two distinct points and of , their distance is certainly positive, so the open balls and are disjoint neighborhoods of and , respectively.

Contractible

A set in which can be reduced to one of its points, say , by a continuous deformation, is said to be contractible. The transformation is such that each point of the set is driven to through a path with the properties that 1. Each path runs entirely inside the set. 2. Nearby points move on "neighboring" paths. Condition (1) implies that a disconnected set,i.e., a set consisting of separate parts, cannot be contractible.Condition (2) implies that the circumference of a circle is not contractible. The latter follows by considering two near points and lying on different sides of a point . The paths connecting and with are either opposite each other or have different lengths. A similar argument shows that, in general, for all , the -sphere (i.e., the boundary of the -dimensional ball) is not contractible.A gap or a hole in a set can be an obstruction to contractibility. There are, however, examples of contractible sets with holes, for example,..

Open cover

A collection of open sets of a topological space whose union contains a given subset. For example, an open cover of the real line, with respect to the Euclidean topology, is the set of all open intervals , where .The set of all intervals , where , is an open cover of the open interval .

Convex function

A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.More generally, a function is convex on an interval if for any two points and in and any where ,(Rudin 1976, p. 101; cf. Gradshteyn and Ryzhik 2000, p. 1132).If has a second derivative in , then a necessary and sufficient condition for it to be convex on that interval is that the second derivative for all in .If the inequality above is strict for all and , then is called strictly convex.Examples of convex functions include for or even , for , and for all . If the sign of the inequality is reversed, the function is called concave.

Open ball

An -dimensional open ball of radius is the collection of points of distance less than from a fixed point in Euclidean -space. Explicitly, the open ball with center and radius is defined byThe open ball for is called an open interval, and the term open disk is sometimes used for and sometimes as a synonym for open ball.

Zariski topology

The Zariski topology is a topology that is well-suited for the study of polynomial equations in algebraic geometry, since a Zariski topology has many fewer open sets than in the usual metric topology. In fact, the only closed sets are the algebraic sets, which are the zeros of polynomials.For example, in , the only nontrivial closed sets are finite collections of points. In , there are also the zeros of polynomials such as lines and cusps .The Zariski topology is not a T2-space. In fact, any two open sets must intersect, and cannot be disjoint. Also, the open sets are dense, in the Zariski topology as well as in the usual metric topology.Because there are fewer open sets than in the usual topology, it is more difficult for a function to be continuous in Zariski topology. For example, a continuous function must be a constant function. Conversely, when the range has the Zariski topology, it is easier for a function to be continuous. In particular, the polynomials..

Convex

A set in Euclidean space is convex set if it contains all the line segments connecting any pair of its points. If the set does not contain all the line segments, it is called concave.A convex set is always star convex, implying pathwise-connected, which in turn implies connected.A region can be tested for convexity in the WolframLanguage using the function Region`ConvexRegionQ[reg].

Ultrametric

An ultrametric is a metric which satisfies the followingstrengthened version of the triangle inequality,for all . At least two of , , and are the same.Let be a set, and let (where N is the set of natural numbers) denote the collection of sequences of elements of (i.e., all the possible sequences , , , ...). For sequences , , let be the number of initial places where the sequences agree, i.e., , , ..., , but . Take if . Then defining gives an ultrametric.The p-adic norm metric is another example ofan ultrametric.

Neighborhood

"Neighborhood" is a word with many different levels of meaning in mathematics.One of the most general concepts of a neighborhood of a point (also called an epsilon-neighborhood or infinitesimal open set) is the set of points inside an -ball with center and radius . A set containing an open neighborhood is also called a neighborhood.The graph neighborhood of a vertex in a graph is the set of all the vertices adjacent to generally including itself. More generally, the th neighborhood of is the set of all vertices that lie at the distance from . The subgraph induced by the neighborhood of a graph from vertex (again, most commonly including itself) is called the neighborhood graph (or sometimes "ego graph" in more recent literature).

Mutually singular

Let be a sigma-algebra , and let and be measures on . If there exists a pair of disjoint sets and such that is concentrated on and is concentrated on , then and are said to be mutually singular, written .

Trivial loop

The trivial loop is the loop that takes every point to its basepoint. Formally, if is a topological space and , the trivial loop based at is the map given by for all .

Multiply connected

A set which is connected but not simply connected is called multiply connected. A space is -multiply connected if it is -connected and if every map from the -sphere into it extends continuously over the -diskA theorem of Whitehead says that a space is infinitelyconnected iff it is contractible.

Topologist's sine curve

An example of a subspace of the Euclidean plane that is connected but not pathwise-connected with respect to the relative topology. It is formed by the ray , and the graph of the function for . This set contains no path connecting the origin with any point on the graph.

Connected space

A space is connected if any two points in can be connected by a curve lying wholly within .A space is 0-connected (a.k.a. pathwise-connected) if every map from a 0-sphere to the space extends continuously to the 1-disk. Since the 0-sphere is the two endpoints of an interval (1-disk), every two points have a path between them. A space is 1-connected (a.k.a. simply connected) if it is 0-connected and if every map from the 1-sphere to it extends continuously to a map from the 2-disk. In other words, every loop in the space is contractible. A space is -multiply connected if it is -connected and if every map from the -sphere into it extends continuously over the -disk.A theorem of Whitehead says that a space is infinitelyconnected iff it is contractible.

Topologically transitive

A function is topologically transitive if, given any two intervals and , there is some positive integer such that . Vaguely, this means that neighborhoods of points eventually get flung out to "big" sets so that they don't necessarily stick together in one localized clump.

Metric space

A metric space is a set with a global distance function (the metric ) that, for every two points in , gives the distance between them as a nonnegative real number . A metric space must also satisfy 1. iff , 2. , 3. The triangle inequality .

Connected set

A connected set is a set that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set. Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other.Let be a topological space. A connected set in is a set which cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set . Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other. The space is a connected topological space if it is a connected subset of itself.The real numbers are a connected set, as are any open or closed interval of real numbers. The (real or complex) plane is connected, as is any open or closed disc or any annulus in the plane. The topologist's sine curve is a connected subset of the plane. An example of..

Topological closure

The closure of a set is the smallest closed set containing . Closed sets are closed under arbitrary intersection, so it is also the intersection of all closed sets containing . Typically, it is just with all of its accumulation points.

Connected im kleinen

The expression im kleinen is German and means "on a small scale." A topological space is connected im kleinen at a point if every neighborhood of contains an open neighborhood of such that any two points of lie in some connected subset of .A locally connected space is connected im kleinen, but the converse is false, ascan be demonstrated by the broom space.

Topological basis

A topological basis is a subset of a set in which all other open sets can be written as unions or finite intersections of . For the real numbers, the set of all open intervals is a basis.Stated another way, if is a set, a basis for a topology on is a collection of subsets of (called basis elements) satisfying the following properties. 1. For each , there is at least one basis element containing . 2. If belongs to the intersection of two basis elements and , then there is a basis element containing such that . (Munkres 2000).

Magic geometric constants

Let be a compact connected subset of -dimensional Euclidean space. Gross (1964) and Stadje (1981) proved that there is a unique real number such that for all , , ..., , there exists with(1)The magic constant of is defined by(2)where(3)These numbers are also called dispersion numbers and rendezvous values. For any , Gross (1964) and Stadje (1981) proved that(4)If is a subinterval of the line and is a circular disk in the plane, then(5)If is a circle, then(6)(OEIS A060294). An expression for the magic constant of an ellipse in terms of its semimajor and semiminor axes lengths is not known. Nikolas and Yost (1988) showed that for a Reuleaux triangle (7)Denote the maximum value of in -dimensional space by . Thenwhere is the gamma function (Nikolas and Yost 1988).An unrelated quantity characteristic of a given magicsquare is also known as a magic constant...

Tietze's extension theorem

A characterization of normal spaces with respect to the definition given by Kelley (1955, p. 112) or Willard (1970, p. 99). It states that the topological space is normal iff, for all closed subsets of , every continuous function , where denotes the real line with the Euclidean topology, can be extended to a continuous function (Willard 1970, p. 103).With respect to the alternative definition (Cullen 1968, p. 118), the statement is different: if is a T4-space, for all closed subsets of , every continuous bounded function can be extended to a continuous bounded function . (Cullen 1968, p. 127)Another characterization of normality in terms of maps is Urysohn'slemma.

Completion

A metric space which is not complete has a Cauchy sequence which does not converge. The completion of is obtained by adding the limits to the Cauchy sequences.For example, the rational numbers, with the distance metric, are not complete because there exist Cauchy sequences that do not converge, e.g., 1, 1.4, 1.41, 1.414, ... does not converge because is not rational. The completion of the rationals is the real numbers. Note that the completion depends on the metric. For instance, for any prime , the rationals have a metric given by the p-adic norm, and then the completion of the rationals is the set of p-adic numbers. Another common example of a completion is the space of L2-functions.Technically speaking, the completion of is the set of Cauchy sequences and is contained in this set, isometrically, as the constant sequences...

Locally connected

A topological space is locally connected at the point if every neighborhood of contains a connected open neighborhood. It is called locally connected if it is locally connected at every point.A connected space need not be locally connected; counterexamples include the comb space and broom space. Conversely, a locally connected space need not be connected; an easy counterexample is the union of two disjoint open intervals of the real line.

Completely regular space

A topological space such that for every closed subset of and every point , there is a continuous function such that and .This is the definition given by most authors (Kelley 1955, p. 117; Willard 1970, pp. 94-95). However, some authors (e.g., Cullen 1968, p. 130) require the additional condition that be a T1-space. In any case, every completely regular space is regular, and the converse is not true.

Locally compact

A topological space is locally compact if every point has a neighborhood which is itself contained in a compact set. Many familiar topological spaces are locally compact, including the Euclidean space. Of course, any compact set is locally compact. Some common spaces are not locally compact, such as infinite dimensional Banach spaces. For instance, the L2-space of square integrable functions is not locally compact.

Subbasis

A collection of subsets of a topological space that is contained in a basis of the topology and can be completed to a basis when adding all finite intersections of the subsets.A subbasis for the Euclidean topology of the real line is formed by all intervals and : in fact a basis is formed by the open intervals .A subbasis for the discrete topology of the real line is formed by all subsets of having a given number of elements, since every singleton set can be obtained as the intersection of the sets and .A subbasis for the Zariski topology of the affine space is formed by the complement sets of all irreducible affine varieties. This follows applying de Morgan's laws when considering that the open sets are the complement sets of the affine varieties, and each of these is the union of a finite number of irreducible varieties...

Subanalytic

is subanalytic if, for all , there is an open set and a bounded semianalytic set such that is the projection of into .

Interval

An interval is a connected portion of the real line. If the endpoints and are finite and are included, the interval is called closed and is denoted . If the endpoints are not included, the interval is called open and denoted . If one endpoint is included but not the other, the interval is denoted or and is called a half-closed (or half-open interval).An interval is called a degenerate interval.If one of the endpoints is , then the interval still contains all of its limit points, so and are also closed intervals. Intervals involving infinity are also called rays or half-lines. If the finite point is included, it is a closed half-line or closed ray. If the finite point is not included, it is an open half-line or open ray.The non-standard notation for an open interval and or for a half-closed interval is sometimes also used.A non-empty subset of is an interval iff, for all and , implies . If the empty set is considered to be an interval, then the following are equivalent:..

Compactification

A compactification of a topological space is a larger space containing which is also compact. The smallest compactification is the one-point compactification. For example, the real line is not compact. It is contained in the circle, which is obtained by adding a point at infinity. Similarly, the plane is compactified by adding one point at infinity, giving the sphere.A topological space has a compactification if and only if it is completely regular and a -space.The extended real line with the order topology is a two point compactification of . The projective plane can be viewed as a compactification of the plane.

Star convex

A subset of is star convex if there exists an such that the line segment from to any point in is contained in .A star-shaped figure is star convex but not convex (as can be seen by taking to be the center of the star.)A star convex set is always pathwise-connected,which in turn is always connected.

Species

A species of structures is a rule which 1. Produces, for each finite set , a finite set , 2. Produces, for each bijection , a function(1)The functions should further satisfy the following functorial properties: 1. For all bijections and ,(2)2. For the identity map ,(3)An element is called an -structure on (or a structure of species on ). The function is called the transport of -structures along .

Simply connected

A pathwise-connected domain is said to be simply connected (also called 1-connected) if any simple closed curve can be shrunk to a point continuously in the set. If the domain is connected but not simply, it is said to be multiply connected. In particular, a bounded subset of is said to be simply connected if both and , where denotes a set difference, are connected.A space is simply connected if it is pathwise-connected and if every map from the 1-sphere to extends continuously to a map from the 2-disk. In other words, every loop in the space is contractible.

Hurwitz's root theorem

Let be a sequence of analytic functions regular in a region , and let this sequence be uniformly convergent in every closed subset of . If the analytic functiondoes not vanish identically, then if is a zero of of order , a neighborhood of and a number exist such that if , has exactly zeros in .

Set closure

A set and a binary operator are said to exhibit closure if applying the binary operator to two elements returns a value which is itself a member of .The closure of a set is the smallest closed set containing . Closed sets are closed under arbitrary intersection, so it is also the intersection of all closed sets containing . Typically, it is just with all of its accumulation points.The term "closure" is also used to refer to a "closed" version of a given set. The closure of a set can be defined in several equivalent ways, including 1. The set plus its limit points, also called "boundary" points, the union of which is also called the "frontier." 2. The unique smallest closed set containing the givenset. 3. The complement of the interior of the complementof the set. 4. The collection of all points such that every neighborhood of these points intersects the original set in a nonempty set. In topologies where the T2-separation..

Separation axioms

A list of five properties of a topological space expressing how rich the "population" of open sets is. More precisely, each of them tells us how tightly a closed subset can be wrapped in an open set. The measure of tightness is the extent to which this envelope can separate the subset from other subsets. The numbering from 0 to 4 refers to an increasing degree of separation.0. T0-separation axiom: For any two points , there is an open set such that and or and . 1. T1-separation axiom: For any two points there exists two open sets and such that and , and and . 2. T2-separation axiom: For any two points there exists two open sets and such that , , and . 3. T3-separation axiom: fulfils and is regular. 4. T4-separation axiom: fulfils and is normal. Some authors (e.g., Cullen 1968, pp. 113 and 118) interchange axiom and regularity, and axiom and normality.A topological space fulfilling is called a -space for short. In the terminology of Alexandroff..

Hereditary property

A property that passes from a topological space to every subspace with respect to the relative topology.Examples are first and second countability, metrizability, the separation axioms , and , and some of the related properties, such as the one of being a regular, completely regular, or Tychonoff space.Axiom is not hereditary, nor is normality, though counterexamples (such as the Tychonoff plank) are hard to find). It is much easier to find disconnected subspaces of connected subspaces (such as, for example, a union of two disjoint disks in the Euclidean plane; left figure) or non-compact subspaces of compact subspaces (e.g., an open disk inside a closed disk; right figure).

Closure

The term "closure" has various meanings in mathematics.The topological closure of a subset of a topological space is the smallest closed subset of containing .If is a binary relation on some set , then has reflexive, symmetric and transitive closures, each of which is the smallest relation on , with the indicated property, containing . Consequently, given any relation on any set , there is always a smallest equivalence relation on containing .For some arbitrary property of relations, the relation need not have a -closure, i.e., there need not be a smallest relation on with the property , and containing . For example, it often happens that a relation does not have an antisymmetric closure.In algebra, the algebraic closure of a field is a field which can be said to be obtained from by adjoining all elements algebraic over ...

First category

A subset of a topological space is said to be of first category in if can be written as the countable union of subsets which are nowhere dense in , i.e., if is expressible as a unionwhere each subset is nowhere dense in . Informally, one thinks of a first category subset as a "small" subset of the host space and indeed, sets of first category are sometimes referred to as meager. Sets which are not of first category are of second category.An important distinction should be made between the above-used notion of "category" and category theory. Indeed, the notions of first and second category sets are independent of category theory.The rational numbers are of first category and the irrational numbers are of second category in with the usual topology. In general, the host space and its topology play a fundamental role in determining category. For example, the set of integers with the subset topology inherited from is (vacuously) of..

Zero set

If is a function on an open set , then the zero set of is the set . A subset of a topological space is called a zero set if it is equal to for some continuous function .

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