General topology

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A homeomorphism, also called a continuous transformation, is an equivalence relation and one-to-one correspondence between points in two geometric figures or topological spaces that is continuous in both directions. A homeomorphism which also preserves distances is called an isometry. Affine transformations are another type of common geometric homeomorphism.The similarity in meaning and form of the words "homomorphism"and "homeomorphism" is unfortunate and a common source of confusion.

Dehn's lemma

An embedding of a 1-sphere in a 3-manifold which exists continuously over the 2-disk also extends over the disk as an embedding. An alternate phrasing is that if a knot group is isomorphic to the group of the integers , then the knot is isomorphic to the unknot (Livingston 1993, p. 104).This theorem was proposed by Dehn in 1910, but a correct proof was not obtained until the work of Papakyriakopoulos (1957ab).

Compactness theorem

Inside a ball in ,is compact under the flat norm.


Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse (into which it can be deformed by stretching) and a sphere is equivalent to an ellipsoid. Similarly, the set of all possible positions of the hour hand of a clock is topologically equivalent to a circle (i.e., a one-dimensional closed curve with no intersections that can be embedded in two-dimensional space), the set of all possible positions of the hour and minute hands taken together is topologically equivalent to the surface of a torus (i.e., a two-dimensional a surface that can be embedded in three-dimensional space), and the set of all possible positions of the hour, minute, and second hands taken together are topologically equivalent to a three-dimensional object.The definition of topology leads to the following mathematical..

Betti number

Betti numbers are topological objects which were proved to be invariants by Poincaré, and used by him to extend the polyhedral formula to higher dimensional spaces. Informally, the Betti number is the maximum number of cuts that can be made without dividing a surface into two separate pieces (Gardner 1984, pp. 9-10). Formally, the th Betti number is the rank of the th homology group of a topological space. The following table gives the Betti number of some common surfaces.surfaceBetti numbercross-cap1cylinder1klein bottle2Möbius strip1plane lamina0projective plane1sphere0torus2Let be the group rank of the homology group of a topological space . For a closed, orientable surface of genus , the Betti numbers are , , and . For a nonorientable surface with cross-caps, the Betti numbers are , , and .The Betti number of a finitely generated Abelian group is the (uniquely determined) number such thatwhere , ..., are finite cyclic..

Euclidean topology

A metric topology induced by the Euclidean metric. In the Euclidean topology of the -dimensional space , the open sets are the unions of -balls. On the real line this means unions of open intervals. The Euclidean topology is also called usual or ordinary topology.


A mathematical property holds locally if is true near every point. In many different areas of mathematics, this notion is very useful. For instance, the sphere, and more generally a manifold, is locally Euclidean. For every point on the sphere, there is a neighborhood which is the same as a piece of Euclidean space.The description of local as "near every point" has a different interpretation in algebra. For instance, given a ring and a prime ideal , there is the local ring , which often is simpler to study. It is possible to understand the original ring better by patching together the information from the local rings.What ties all the notions of local together is the concept of a topology, a collection of open sets. For a submanifold of Euclidean space, or for the set of ideals of a ring, the topology is chosen as is appropriate.A property holds locally on a topological space if every point has a neighborhood on which holds. This concept is useful..

Urysohn's metrization theorem

For every topological T1-space , the following conditions are equivalent. 1. is regular and second countable, 2. is separable and metrizable. 3. is homeomorphic to a subspace of the Hilbert cube.


The dimension of an object is a topological measure of the size of its covering properties. Roughly speaking, it is the number of coordinates needed to specify a point on the object. For example, a rectangle is two-dimensional, while a cube is three-dimensional. The dimension of an object is sometimes also called its "dimensionality."The prefix "hyper-" is usually used to refer to the four- (and higher-) dimensional analogs of three-dimensional objects, e.g., hypercube, hyperplane.The notion of dimension is important in mathematics because it gives a precise parameterization of the conceptual or visual complexity of any geometric object. In fact, the concept can even be applied to abstract objects which cannot be directly visualized. For example, the notion of time can be considered as one-dimensional, since it can be thought of as consisting of only "now," "before" and "after."..

Urysohn's lemma

A characterization of normal spaces which states that a topological space is normal iff, for any two nonempty closed disjoint subsets , and of , there is a continuous map such that and . A function with this property is called a Urysohn function.This formulation refers to the definition of normal space given by Kelley (1955, p. 112) or Willard (1970, p. 99). In the statement for an alternative definition (e.g., Cullen 1968, p. 118), the word "normal" has to be replaced by .

Tychonoff theorem

A product space is compact iff is compact for all . In other words, the topological product of any number of compact spaces is compact. In particular, compactness is a productive property. As a consequence, every Hilbert cube is compact.This statement implies the axiom of choice, asproven by Kelley (1950).


A topologically invariant property of a surface defined as the largest number of nonintersecting simple closed curves that can be drawn on the surface without separating it. Roughly speaking, it is the number of holes in a surface.The genus of a surface, also called the geometric genus, is related to the Euler characteristic . For a orientable surface such as a sphere (genus 0) or torus (genus 1), the relationship isFor a nonorientable surface such as a real projective plane (genus 1) or Klein bottle (genus 2), the relationship is(Massey 2003).


A sheaf is a presheaf with "something" added allowing us to define things locally. This task is forbidden for presheaves in general. Specifically, a presheaf on a topological space is a sheaf if it satisfies the following conditions: 1. if is an open set, if is an open covering of and if is an element such that for all , then . 2. if is an open set, if is an open covering of and if we have elements for each , with the property that for each, , , then there is an element such that for all . The first condition implies that is unique.For example, let be a variety over a field . If denotes the ring of regular functions from to then with the usual restrictions is a sheaf which is called the sheaf of regular functions on .In the same way, one can define the sheaf of continuous real-valued functions on any topological space, and also for differentiable functions...

Euler characteristic

Let a closed surface have genus . Then the polyhedral formula generalizes to the Poincaré formula(1)where(2)is the Euler characteristic, sometimes also known as the Euler-Poincaré characteristic. The polyhedral formula corresponds to the special case .The only compact closed surfaces with Euler characteristic 0 are the Klein bottle and torus (Dodson and Parker 1997, p. 125). The following table gives the Euler characteristics for some common surfaces (Henle 1994, pp. 167 and 295; Alexandroff 1998, p. 99).surfacecylinder0double torusKlein bottle0Möbius strip0projective plane1sphere2torus0In terms of the integral curvature of the surface ,(3)The Euler characteristic is sometimes also called the Eulernumber. It can also be expressed as(4)where is the th Betti number of the space...

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