In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent...

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...

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...

} (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...

compact space is limit point compact. For T1 spaces, countable compactness and limit point compactness are equivalent. Every sequentially compact space is...

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...

which the sequential closure operator is defined, the topological space ( X , T ) {\displaystyle (X,{\mathcal {T}})} is a sequential space if and only...

a metric space, but may not be equivalent in other topological spaces. One such generalization is that a topological space is sequentially compact if...

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...

(this shortcoming of sequences motivated the study of sequential spaces and Fréchet–Urysohn spaces). Nets are in one-to-one correspondence with filters...

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...

In statistics, sequential analysis or sequential hypothesis testing is statistical analysis where the sample size is not fixed in advance. Instead data...

spaces are called US spaces. For sequential spaces, this notion is equivalent to being weakly hausdorff. Subspaces and products of Hausdorff spaces are...

topological spaces include: sequential space: a set is open if every sequence convergent to a point in the set is eventually in the set first-countable space: every...

topological spaces whose definitions all somehow relate sequences in the codomain with sequences in the domain. Examples include sequentially quotient maps...

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...

topology of a sequential space (or a Fréchet space) is determined by the convergent sequences. The countably generated spaces are precisely the spaces having...

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...

locally convex topological vector space (such as any normed space) is continuous if and only if it is sequentially continuous at the origin. However,...

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...

\mathbb {R} ^{n}} is sequentially compact if and only if it is closed and bounded. The theorem is sometimes called the sequential compactness theorem....

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...

{\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...

National Aeronautics and Space Administration (NASA) as part of the Space Shuttle program. Its official program name was Space Transportation System (STS)...

space of a DF-space is a Fréchet space. Every infinite-dimensional Montel DF-space is a sequential space but not a Fréchet–Urysohn space. Suppose X {\displaystyle...

U ) {\displaystyle {\mathcal {D}}^{\prime }(U)} is a sequential space (not even an Ascoli space), which in particular implies that their topologies can...

{\displaystyle X} is a metric space, sequential continuity and continuity are equivalent. For non-first-countable spaces, sequential continuity might be strictly...

In statistics, sequential estimation refers to estimation methods in sequential analysis where the sample size is not fixed in advance. Instead, data is...

Particle filters, or sequential Monte Carlo methods, are a set of Monte Carlo algorithms used to find approximate solutions for filtering problems for...

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...