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 map $p$ of topological spaces is called a Serre microfibration if for any lifting square for $\{0\}\times K\to [0,1]\times K$ and $p$, we can find $\epsilon\gt0$ such that the lifting property is satisfied after restricting to $[0,\epsilon]\times K\subset [0,1]\times K$.
Any Serre fibration is a Serre microfibration.
Any inclusion of open subspaces is a Serre microfibration. It is a Serre fibration if and only it is a homeomorphism.
Last revised on November 6, 2021 at 06:41:46. See the history of this page for a list of all contributions to it.