Countably generated space
Encyclopedia
In mathematics
, a topological space
X is called countably generated if the topology of X is determined by the countable sets in a similar way as the topology of a sequential space
(or a Fréchet space
) by the convergent sequences.
The countable generated spaces are precisely the spaces having countable tightness - therefore the name countably tight is used as well.
is closed in U. Equivalently, X is countably generated if and only if the closure of any subset A of X equals the union of closures of all countable subsets of A.
A quotient
of countably generated space is again countably generated. Similarly, a topological sum of countably generated spaces is countably generated. Therefore the countably generated spaces form a coreflective subcategory of the category of topological spaces
. They are the coreflective hull of all countable spaces.
Any subspace of a countably generated space is again 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X is called countably generated if the topology of X is determined by the countable sets in a similar way as the topology of a sequential space
Sequential space
In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability. Sequential spaces are the most general class of spaces for which sequences suffice to determine the topology....
(or a Fréchet space
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...
) by the convergent sequences.
The countable generated spaces are precisely the spaces having countable tightness - therefore the name countably tight is used as well.
Definition
A topological space X is called countably generated if V is closed in X whenever for each countable subspace U of X the set is closed in U. Equivalently, X is countably generated if and only if the closure of any subset A of X equals the union of closures of all countable subsets of A.A quotient
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
of countably generated space is again countably generated. Similarly, a topological sum of countably generated spaces is countably generated. Therefore the countably generated spaces form a coreflective subcategory of the category of topological spaces
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 continuous...
. 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) 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.
External links
- A Glossary of Definitions from General Topology http://math.berkeley.edu/~apollo/topodefs.ps
- http://thales.doa.fmph.uniba.sk/density/pages/slides/sleziak/paper.pdf