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Alexandroff plank in topology, an area of mathematics, is a topological space that serves as an instructive example.
The construction of the Alexandroff plank starts by defining the topological space to be the Cartesian product of and , where is the first uncountable ordinal, and both carry the interval topology. The topology is extended to a topology by adding the sets of the form
where .
The Alexandroff plank is the topological space .
It is called plank for being constructed from a subspace of the product of two spaces.
The space satisfies that:
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a real-valued function defined on the vector space that is commonly denoted and has the following properties:
In mathematical analysis and in probability theory, a σ-algebra on a set X is a collection Σ of subsets of X, is closed under complement, and is closed under countable unions and countable intersections. The pair is called a measurable space.
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.
In mathematics, a base or basis for the topology τ of a topological space (X, τ) is a family B of open subsets of X such that every open set of the topology is equal to a union of some sub-family of B. For example, the set of all open intervals in the real number line is a basis for the Euclidean topology on because every open interval is an open set, and also every open subset of can be written as a union of some family of open intervals.
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.
In mathematics, a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures.
In mathematics, a filtration is an indexed family of subobjects of a given algebraic structure , with the index running over some totally ordered index set , subject to the condition that
In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relevance to proof theory still have computable ordinal notations. However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not ; various more-concrete ways of defining ordinals that definitely have notations are available.
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of countability. All first-countable spaces, which includes metric spaces, are sequential spaces.
In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called the rank of the Borel set. The Borel hierarchy is of particular interest in descriptive set theory.
In mathematics, the first uncountable ordinal, traditionally denoted by or sometimes by , is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum of all countable ordinals. When considered as a set, the elements of are the countable ordinals, of which there are uncountably many.
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals aimed to extend enumeration to infinite sets.
In the fields of computer vision and image analysis, the scale-invariant feature operator is an algorithm to detect local features in images. The algorithm was published by Förstner et al. in 2009.
In mathematics, the Gevrey classes on a domain , introduced by Maurice Gevrey, are spaces of functions 'between' the space of analytic functions and the space of smooth functions . In particular, for , the Gevrey class , consists of those smooth functions such that for every compact subset there exists a constant , depending only on , such that
In mathematics, a polyadic space is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space.
In Category theory and related fields of mathematics, an envelope is a construction that generalizes the operations of "exterior completion", like completion of a locally convex space, or Stone–Čech compactification of a topological space. A dual construction is called refinement.
In mathematical logic, the intersection type discipline is a branch of type theory encompassing type systems that use the intersection type constructor to assign multiple types to a single term. In particular, if a term can be assigned both' the type and the type , then can be assigned the intersection type . Therefore, the intersection type constructor can be used to express finite heterogeneous ad hoc polymorphism . For example, the λ-term can be assigned the type in most intersection type systems, assuming for the term variable both the function type and the corresponding argument type .
In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS Y without any need to extend definitions from real/complex-valued functions to Y-valued functions.
In mathematical logic, the Buchholz hydra game is a hydra game, which is a single player game based on the idea of chopping pieces off a mathematical tree. The hydra game can be used to generate a rapidly growing function , which eventually dominates all recursive functions that are provably total in , and is itself provably total in + "transfinite induction with respect to TFB".
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