topological spaces. They are the coreflective hull of all countable spaces. Any subspace of a countably generated space is again countably generated. Every...

generating set Countably generated space, a topological space in which the topology is determined by its countable subsets Countably generated module. (Kaplansky's...

first-countable space is compactly generated. Every subspace of a first-countable space is first-countable. Any countable product of a first-countable space...

a topological space X {\displaystyle X} is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made...

to some Borel set) but not countably generated (since its cardinality is higher than continuum). A separable measure space has a natural pseudometric...

Fréchet space. A product of countably many Fréchet spaces is always once again a Fréchet space. However, an arbitrary product of Fréchet spaces will be...

countably many locally finite collections of open sets. For a closely related theorem see the Bing metrization theorem. Separable metrizable spaces can...

In mathematics, a topological space X {\displaystyle X} is said to be limit point compact or weakly countably compact if every infinite subset of X {\displaystyle...

countably additive (also called σ-additive): if { A i } i = 1 ∞ ⊆ F {\displaystyle \{A_{i}\}_{i=1}^{\infty }\subseteq {\mathcal {F}}} is a countable collection...

countability, which is in general stronger but equivalent on the class of metrizable spaces. Any topological space that is itself finite or countably...

is countably infinite, it allows identifying the Hilbert space with the space of the infinite sequences that are square-summable. The latter space is...

topological space is said to be submaximal if every subset is locally closed. See Glossary of topology#S for more of this notion. Countably generated space – topological...

"Absolutely measurable" means: measurable in every countably separated, countably generated probability space containing it. Definition.   ( Ω , F , P ) {\displaystyle...

simply connected after removal of countably many points. Any topological vector space, e.g. any Hilbert space or Banach space, over a connected field (such...

original topological space (a disjoint union of countably many intervals) lead to different topologies on the quotient. A topological space is sequential if...

equivalent to compactness for first-countable uniform spaces). (X, d) is limit point compact (also called weakly countably compact); that is, every infinite...

ω1 is a countable set; for any sequence of such sets, the union of these sets is the union of countably many countable sets, so still countable; this union...

Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense...

Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis...

connected, locally connected and pseudocompact, but neither weakly countably compact nor countably metacompact, hence not compact. Uncountable set: On any uncountable...

are countably infinite-dimensional vector spaces, and many function spaces have the cardinality of the continuum as a dimension. Many vector spaces that...

very weak axiom of countability, and all first-countable spaces (notably metric spaces) are sequential. In any topological space ( X , τ ) , {\displaystyle...

is pseudocompact and weakly countably compact. Countably locally finite A collection of subsets of a space X is countably locally finite (or σ-locally...

the topology generated by Γ, which is the Euclidean topology on R {\displaystyle \mathbb {R} } , is coarser than the topology generated by Σ. In fact...

finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional vector space, and...

group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not...

compact. Quotient spaces of locally compact Hausdorff spaces are compactly generated. Conversely, every compactly generated Hausdorff space is a quotient...

all non-empty subsets of a topological space X , {\displaystyle X,} named for Leopold Vietoris, is generated by the following basis: for every n {\displaystyle...

mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an...

A collection in a space X {\displaystyle X} is countably locally finite (or 𝜎-locally finite) if it is the union of a countable family of locally finite...