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
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
A simplicial topological space is a simplicial object in Top.
Often this is called just a simplicial space , if the context is clear.
A special case is that of simplicial manifolds.
Often one is interested in simplicial topological spaces with extra nice properties. See nice simplicial topological space for more on that.
As with simplicial objects in general, simplicial spaces may serve to model internal ∞-groupoids in Top. Notably there is a rich theory of simplicial topological groups.
A standard textbook reference is chapter 11 of
Last revised on June 25, 2021 at 14:09:09. See the history of this page for a list of all contributions to it.