Topology Topology An adjunction space is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be a topological spaces with A a subspace of Y. Let f : A → X be continuous map (called the attaching map). One forms the adjunction space X ∪f Y by taking the disjoint union of X and Y and identifying x ∼ f(x) for all x in A. Schematically, ...more on Wikipedia about "Adjunction space"
An affine representation of a topological ( Lie) group G is a continuous ( smooth) homomorphism from G to the automorphism group of an affine space, A. ...more on Wikipedia about "Affine representation"
The Alexander horned sphere is one of the most famous pathological examples in mathematics. ...more on Wikipedia about "Alexander horned sphere"
The obvious asymmetry in these conditions leads one to ask: "What happens when the intersection of arbitrarily many open sets is open?" The answer is, the Alexandrov topology. ...more on Wikipedia about "Alexandrov topology"
In mathematics, an algebra bundle is a fiber bundle whose fibers are algebras and matching functions are algebra isomorphisms. ...more on Wikipedia about "Algebra bundle"
Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...more on Wikipedia about "Algebraic topology"
Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular computational complexity theory. ...more on Wikipedia about "Algorithmic topology"
Antipodal points on the surface of a sphere are diametrically opposite - so situated that a line drawn from the one to the other passes through the centre of the globe and forms a true diameter. For example, "Spain and New Zealand lie in antipodal regions." ...more on Wikipedia about "Antipodal point"
In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general. ...more on Wikipedia about "Ball (mathematics)"
The Banach fixed point theorem is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive method to find those fixed points. The theorem is named after Stefan Banach (1892-1945), and was first stated by Banach in 1922. ...more on Wikipedia about "Banach fixed point theorem"
In geometric topology, a band sum of two -dimensional knots and along an -dimensional 1-handle called a band is an -dimensional knot such that: ...more on Wikipedia about "Band sum"
In mathematics, the Bernoulli numbers are a series of rational numbers with deep connections in number theory. Although easy to calculate, the values of the Bernoulli numbers have no elementary description; they are, up to a factor, the values of the Riemann zeta function at negative integers. ...more on Wikipedia about "Bernoulli number"
In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space X is a σ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this σ-algebra: ...more on Wikipedia about "Borel algebra"
In topology a boundary component of a compact surface is a connected component consisting of boundary points of a surface. A point in any surface is either a boundary point, if it has a neighborhood homeomorphic to a half-disk, or an interior point, if it has a neighborhood homeomorphic to a disk. ...more on Wikipedia about "Boundary component"
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In mathematics, the Brouwer fixed point theorem states that every continuous function from the closed unit ball D n to itself has a fixed point. In this theorem, n is any positive integer, and the closed unit ball is the set of all points in Euclidean n-space Rn which are at distance at most 1 from the origin. Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, the theorem equally applies if the domain is not the closed unit ball itself but some set homeomorphic to it (and therefore also closed, connected, without holes, etcetera). ...more on Wikipedia about "Brouwer fixed point theorem"
In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. To be more precise, by dropping a finite number of elements from the start of the sequence we can make the distance between any two remaining elements arbitrarily small. ...more on Wikipedia about "Cauchy sequence"
In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set. In other words, Y contains all but countably many elements of X. ...more on Wikipedia about "Cocountable"
In mathematics, a cofinite subset of a set X is a subset Y whose complement in X is a finite set. In other words, Y contains all but finitely many elements of X. ...more on Wikipedia about "Cofinite"
In mathematics, a coincidence point (or simply coincidence) of two mappings is a point in their domain having the same image point under both mappings. ...more on Wikipedia about "Coincidence point"
In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. For example, in R, the closed unit interval [0, 1] is compact, but the set of integers Z is not (it is not bounded) and neither is the half-open interval
In mathematics, compactification is applied to topological spaces to make them compact spaces. There is no unique way to do this. ...more on Wikipedia about "Compactification (mathematics)" Fast shortopedia
In topology, especially algebraic topology, the cone CX of a topological space X is the quotient space: ...more on Wikipedia about "Cone (topology)"
A containment hierarchy is a hierarchical collection of strictly nested sets. Each entry in the hierarchy designates a set such that the previous entry is a strict superset, and the next entry is a strict subset. For example, all rectangles are quadrilaterals, but not all quadrilaterals are rectangles, and all squares are rectangles, but not all rectangles are squares. ...more on Wikipedia about "Containment hierarchy"
In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point. A contractible space is precisely one with the homotopy type of a point. ...more on Wikipedia about "Contractible space"
In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if ...more on Wikipedia about "Cover (topology)"
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