In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent...
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In mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in X...
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identical to the sequential closure of S {\displaystyle S} in X . {\displaystyle X.} Fréchet–Urysohn spaces are a special type of sequential space. The property...
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} (In other words, every first-countable space is a Fréchet-Urysohn space and thus also a sequential space.) This has consequences for limits and continuity...
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the term T1 space is preferred. There is also a notion of a Fréchet–Urysohn space as a type of sequential space. The term symmetric space also has another...
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after E. G. Pytkeev, who proved in 1983 that sequential spaces have this property. Let X be a topological space. For a subset S of X let S denote the closure...
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spaces are called US spaces. For sequential spaces, this notion is equivalent to being weakly Hausdorff. Subspaces and products of Hausdorff spaces are...
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which is contained in the set. Every premetric space is a topological space, and in fact a sequential space. In general, the r {\displaystyle r} -balls themselves...
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Net (mathematics) (redirect from Net (of sets in a topological space))
(this shortcoming of sequences motivated the study of sequential spaces and Fréchet–Urysohn spaces). Nets are in one-to-one correspondence with filters...
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topology of a sequential space (or a Fréchet space) is determined by the convergent sequences. The countably generated spaces are precisely the spaces having...
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(reasonable) parallel machine is polynomially related to the space used by a sequential machine. The parallel computation thesis was set forth by Chandra...
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uniform space X is said to be sequentially complete or semi-complete if every Cauchy sequence in S converges to an element in S. X is called sequentially complete...
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General topology (redirect from Point set space)
σ-compact space: there exists a countable cover by compact spaces Relations: Every first countable space is sequential. Every second-countable space is first-countable...
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a metric space, but may not be equivalent in other topological spaces. One such generalization is that a topological space is sequentially compact if...
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Preclosure operator (section Sequential spaces)
which the sequential closure operator is defined, the topological space ( X , T ) {\displaystyle (X,{\mathcal {T}})} is a sequential space if and only...
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provably exist. Important countability axioms for topological spaces include: sequential space: a set is closed if and only if every convergent sequence in...
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compact space is limit point compact. For T1 spaces, countable compactness and limit point compactness are equivalent. Every sequentially compact space is...
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{\displaystyle X} is a sequential space (such as a pseudometrizable space) then this list may be extended to include: F {\displaystyle F} is sequentially continuous...
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Particle filter (redirect from Sequential Monte Carlo method)
Particle filters, also known as sequential Monte Carlo methods, are a set of Monte Carlo algorithms used to find approximate solutions for filtering problems...
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Sequence covering map (redirect from Sequentially quotient)
topological spaces whose definitions all somehow relate sequences in the codomain with sequences in the domain. Examples include sequentially quotient maps...
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only if it is continuous. For LF spaces, a weaker converse holds; any bounded linear map from an LF space is sequentially continuous. If F : X → Y {\displaystyle...
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Distribution (mathematics) (redirect from Distribution space)
locally convex topological vector space (such as any normed space) is continuous if and only if it is sequentially continuous at the origin. However,...
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equivalent. Sequential spaces are CG-2. This includes first countable spaces, Alexandrov-discrete spaces, finite spaces. Every CG-3 space is a T1 space (because...
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Whitespace character (redirect from White-space character)
right-to-left script) or to the start of the next line. The effect of multiple sequential whitespace characters is cumulative such that the next printable character...
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Continuous function (redirect from Sequential continuity)
{\displaystyle X} is a metric space, sequential continuity and continuity are equivalent. For non-first-countable spaces, sequential continuity might be strictly...
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Bolzano–Weierstrass theorem (redirect from Sequential compactness theorem)
\mathbb {R} ^{n}} is sequentially compact if and only if it is closed and bounded. The theorem is sometimes called the sequential compactness theorem....
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ISAM (redirect from Indexed Sequential Access Method)
Indexed Sequential Access Method (ISAM) is a method for creating, maintaining, and manipulating computer files of data so that records can be retrieved...
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In statistics, sequential analysis or sequential hypothesis testing is statistical analysis where the sample size is not fixed in advance. Instead data...
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infinite-dimensional spaces, a set that is closed and bounded is not necessarily (sequentially) compact (as is the case in all finite dimensional spaces). Indeed...
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Paracompact space Locally compact space Compactly generated space Axiom of countability Sequential space First-countable space Second-countable space Separable...
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