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...

mathematics, compactly generated can refer to: Compactly generated group, a topological group which is algebraically generated by one of its compact subsets...

cube is compact, again a consequence of Tychonoff's theorem. A profinite group (e.g. Galois group) is compact. Compactly generated space Compactness theorem...

mathematics, a compactly generated (topological) group is a topological group G which is algebraically generated by one of its compact subsets. This should...

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

with compactly generated spaces in algebraic topology. For that, see the category of compactly generated weak Hausdorff spaces. A k-Hausdorff space is a...

of compactly generated weak Hausdorff spaces, CGWH, is a category used in algebraic topology as an alternative to the category of topological spaces, Top...

for a compactly generated space in topology K-space (functional analysis) is an F-space such that every twisted sum by the real line splits K-Space (band)...

with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak Hausdorff spaces. Like many...

coincides with the compact-open topology and may be used to uniquely define it. The modification of the definition for compactly generated spaces may be viewed...

{\displaystyle (X,{\mathcal {T}})} is a compactly generated space, f n → f {\displaystyle f_{n}\to f} compactly, and each f n {\displaystyle f_{n}} is...

"convenient". Every sequential space is compactly generated, and finite products in Seq coincide with those for compactly generated spaces, since products in the...

difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formally, a homotopy...

of non-compactness Paracompact space Locally compact space Compactly generated space Axiom of countability Sequential space First-countable space Second-countable...

here. As an example, a commonly used variant of the notion of compactly generated space is defined as the final topology with respect to a proper class...

the change in functions More generally, on any compactly generated space; e.g., a first-countable space. Rudin 1991, p. 44 §2.5. Reed & Simon (1980), p...

containing all the others. It is the algebra generated by the product topology or weak topology on the product space { H , T } ∞ . {\displaystyle \{H,T\}^{\infty...

that X and Y are compactly generated Hausdorff spaces, so Hom(X,Y) is often taken with the compactly generated variant of the compact-open topology; the...

topology, i.e., the topology generated by all intervals ( x , ∞ ) . {\displaystyle (x,\infty ).} The space is limit point compact because given any point a...

Banach space ℓ 1 {\displaystyle \ell ^{1}} is not a K-space. Compactly generated space – Property of topological spaces Gelfand–Shilov space Kalton,...

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

non-compact, but this is not an open manifold since the circle (one of its components) is compact. Most books generally define a manifold as a space that...

D({\text{Sh}}(X;{\text{Ab}}))} for a non-compact topological space X {\displaystyle X} , it is generally not a compactly generated category. Some evidence for this...

locally compact Polish space X {\displaystyle X} is a variant of the Vietoris topology, and is named after mathematician James Fell. It is generated by the...

arguments). For any pointed spaces X, Y, and Z in an appropriate "convenient" category (e.g., that of compactly generated spaces), there are natural (basepoint...

{\displaystyle g\colon A\to B} is any (continuous) map (between compactly generated spaces), and i : A → X {\displaystyle i\colon A\to X} is a cofibration...

metric space. More general formulations of the theorem exist that give necessary and sufficient conditions for a family of functions from a compactly generated...

The compact elements of Sub(A) are exactly the finitely generated substructures of A. Every substructure is the union of its finitely generated substructures;...

product and R as the unit. The category of pointed spaces (restricted to compactly generated spaces for example) is monoidal with the smash product serving...

preorder on the space. Spaces with an Alexandrov topology are also known as Alexandrov-discrete spaces or finitely generated spaces. The latter name...