topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
The classical model structure on topological spaces $Top_{Qu}$ restricts to compactly generated topological spaces and further to weak Hausdorff spaces among these, to yield Quillen equivalent model structures $k Top_{Qu}$ and $h k Top_{Qu}$there. Since these model structures on k-spaces are Cartesian monoidal closed as model categories (Prop. below) while $Top_{QU}$ is not, they provide a more convenient foundations for much of homotopy theory in terms of model categories.
Recall (from here) the sequence of adjoint functors
exhibiting the coreflective subcategory inside all of Top of compactly generated topological spaces and further the reflective subcategory of weak Hausdorff spaces among these.
(classical model structure on compactly generated topological spaces)
The classical model structure on topological spaces $Top_{Qu}$ restricts along $k Top \xhookrightarrow{\;} Top$ (1) to a cofibrantly generated model category structure $k Top_{Qu}$ on compactly generated topological spaces, and the coreflection becomes a Quillen equivalence:
The model structure on compactly generated topological spaces from Prop. is a cartesian monoidal model category;
Similarly:
(classical model structure on compactly generated weak Hausdorff spaces)
The model structure on compactly generated topological spaces $k Top_{Qu}$ from Prop. restricts along $h k top \xhookrightarrow{\;} k Top$ (1) to a model category structure on weakly Hausdorff k-spaces $h k Top_{Qu}$, and the reflection is a Quillen equivalence:
Again, this is a Cartesian monoidal model category (e.g. Hovey 1999, Thm. 2.4.23).
In the same way as above (e.g. Gaucher 2007, p. 7, Haraguchi 2013):
the model structure on compactly generated topological spaces restricts further along the inclusion of Delta-generated topological spaces $D Top \hookrightarrow k Top$, to give a Quillen equivalent model structure on Delta-generated topological spaces:
Textbook accounts:
Lecture notes:
See also:
Last revised on September 30, 2021 at 07:19:08. See the history of this page for a list of all contributions to it.