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, and all first-countable spaces (especially metric spaces) are sequential.
In any topological space if a convergent sequence is contained in a closed set then the limit of that sequence must be contained in as well. Sets with this property are known as sequentially closed. Sequential spaces are precisely those topological spaces for which sequentially closed sets are in fact closed. (These definitions can also be rephrased in terms of sequentially open sets; see below.) Said differently, any topology can be described in terms of nets (also known as Moore–Smith sequences), but those sequences may be "too long" (indexed by too large an ordinal) to compress into a sequence. Sequential spaces are those topological spaces for which nets of countable length (i.e., sequences) suffice to describe the topology.
Any topology can be refined (that is, made finer) to a sequential topology, called the sequential coreflection of
The related concepts of Fréchet–Urysohn spaces, T-sequential spaces, and -sequential spaces are also defined in terms of how a space's topology interacts with sequences, but have subtly different properties.
Sequential spaces and -sequential spaces were introduced by S. P. Franklin.[1]
^Cite error: The named reference Snipes T-sequential spaces was invoked but never defined (see the help page).
In topology and related fields of mathematics, a sequentialspace 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...
compact space is limit point compact. For T1 spaces, countable compactness and limit point compactness are equivalent. Every sequentially compact space is...
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 sequentialspace. In general, the r {\displaystyle r} -balls themselves...
which the sequential closure operator is defined, the topological space ( X , T ) {\displaystyle (X,{\mathcal {T}})} is a sequentialspace if and only...
the term T1 space is preferred. There is also a notion of a Fréchet–Urysohn space as a type of sequentialspace. The term symmetric space also has another...
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 sequentialspaces, this notion is equivalent to being weakly hausdorff. Subspaces and products of Hausdorff spaces are...
uniform space ( X , U ) {\displaystyle (X,{\mathcal {U}})} is called a complete uniform space (respectively, a sequentially complete uniform space) if every...
topology of a sequentialspace (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...
topological spaces include: sequentialspace: a set is open if every sequence convergent to a point in the set is eventually in the set first-countable space: every...
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...
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...
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...
{\displaystyle X} is a metric space, sequential continuity and continuity are equivalent. For non-first-countable spaces, sequential continuity might be strictly...
U ) {\displaystyle {\mathcal {D}}^{\prime }(U)} is a sequentialspace (not even an Ascoli space), which in particular implies that their topologies can...
{\displaystyle X} is a sequentialspace (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)...
after E. G. Pytkeev, who proved in 1983 that sequentialspaces have this property. Let X be a topological space. For a subset S of X let S denote the closure...
equivalent. Sequentialspaces are CG-2. This includes first countable spaces, Alexandrov-discrete spaces, finite spaces. Every CG-3 space is a T1 space (because...
In statistics, sequential estimation refers to estimation methods in sequential analysis where the sample size is not fixed in advance. Instead, data is...
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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