HOME

TheInfoList



Click Here for Items Related To - Countably Generated Space
OR:
Countably Generated Space on:   [Wikipedia]   [Google]   [Amazon]

In mathematics, a
topological 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 whose elements are called po ...
X is called countably generated if the topology of X is determined by the
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
sets in a similar way as the topology of a
sequential space 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 count ...
(or a
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect ...
) is determined by the convergent sequences. The countably generated spaces are precisely the spaces having countable
tightness Tightness may refer to: In mathematics, * Tightness of (a collection of) measures is a concept in measure (and probability) theory * Tightness (topology) is also a cardinal function used in general topology In economics, * Tightness refers t ...
—therefore the name ' is used as well.


Definition

A topological space X is called if for every subset V \subseteq X, V is closed in X whenever for each countable
subspace Subspace may refer to: In mathematics * A space inheriting all characteristics of a parent space * A subset of a topological space endowed with the subspace topology * Linear subspace, in linear algebra, a subset of a vector space that is closed ...
U of X the set V \cap U is closed in U. Equivalently, X is countably generated if and only if the closure of any A \subseteq X equals the union of closures of all countable subsets of A.


Countable fan tightness

A topological space X has if for every point x \in X and every sequence A_1, A_2, \ldots of subsets of the space X such that x \in \, \overline = \overline \cap \overline \cap \cdots, there are finite set B_1\subseteq A_1, B_2 \subseteq A_2, \ldots such that x \in \overline = \overline. A topological space X has if for every point x \in X and every sequence A_1, A_2, \ldots of subsets of the space X such that x \in \, \overline = \overline \cap \overline \cap \cdots, there are points x_1 \in A_1, x_2 \in A_2, \ldots such that x \in \overline. Every strong Fréchet–Urysohn space has strong countable fan tightness.


Properties

A
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of a countably generated space is again countably generated. Similarly, a
topological sum In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of ...
of countably generated spaces is countably generated. Therefore the countably generated spaces form a
coreflective subcategory In mathematics, a full subcategory ''A'' of a category ''B'' is said to be reflective in ''B'' when the inclusion functor from ''A'' to ''B'' has a left adjoint. This adjoint is sometimes called a ''reflector'', or ''localization''. Dually, ''A ...
of the
category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ...
. They are the coreflective hull of all countable spaces. Any subspace of a countably generated space is again countably generated.


Examples

Every sequential space (in particular, every
metrizable space In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty ...
) is countably generated. An example of a space which is countably generated but not sequential can be obtained, for instance, as a subspace of
Arens–Fort space In mathematics, the Arens–Fort space is a special example in the theory of topological spaces, named for Richard Friederich Arens and M. K. Fort, Jr. Definition The Arens–Fort space is the topological space (X,\tau) where X is the set of orde ...
.


See also

* * * − Tightness is a cardinal function related to countably generated spaces and their generalizations.


References

*


External links

* A Glossary of Definitions from General Topolog

* https://web.archive.org/web/20040917084107/http://thales.doa.fmph.uniba.sk/density/pages/slides/sleziak/paper.pdf General topology {{topology-stub