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...

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)...

-Hausdorff spaces are to Δ {\displaystyle \Delta } -generated spaces as weak Hausdorff spaces are to compactly generated spaces. Fixed-point space – Space where...

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...

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...

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

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...

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...

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...

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...

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...

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...

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...

hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex...

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

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...

Moreover, a (Hausdorff) space is realcompact if and only if it has the uniform topology and is complete for the uniform structure generated by the continuous...

{\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...

For any indexed family of topological spaces, the product can be given the product topology, which is generated by the inverse images of open sets of...

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