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
The profinite completion of the integers is the inverse limit
of all the cyclic groups over their canonical filtered diagram.
This is isomorphic to the product of the p-adic integers for all prime numbers $p$
From this perspective the concept of the ring of adeles is natural, see there for more.
Under Pontryagin duality, $\hat \mathbb{Z}$ maps to $\mathbb{Q}/\mathbb{Z}$, see at Pontryagin duality for torsion abelian groups.
Hendrik Lenstra, example 2.2 in Profinite groups (pdf)
Last revised on September 2, 2021 at 08:39:14. See the history of this page for a list of all contributions to it.