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 forgetful functor/full and faithful subcategory embedding from compact Hausdorff topological spaces into all topological spaces
has a left adjoint
which sends a general topological space to a compact Hausdorff topological space, called its Stone-Čech compactification. This hence exhibits $Top_{CHaus}$ as a reflective subcategory of all of $Top$.
The Stone-Cech compactification is in general “very large”, even for “ordinary” non-compact spaces such as the real line.
For more details see at compactum – Stone-Čech compactification
The unit
of the compactification adjunction $(\beta \dashv U)$ is an embedding precisely for $X$ a Tychonoff space.
The Stone-Cech compactification of a discrete topological space is an extremally disconnected topological space. By a theorem by Gleason, these are precisely the projective objects in the category of compact Hausdorff topological spaces.
Such spaces appear for instance as connected components of w-contractible rings as objects in the pro-étale site. See (Bhatt-Scholze 13).
Lecture notes include
Tarun Chitra, The Stone-Cech Compactification 2009 pdf
Brian Bockelman, Functional Analysis Notes, The Stone-Cech compactification
Discussion in the context of the pro-etale site is in
See also
Last revised on June 13, 2021 at 14:58:06. See the history of this page for a list of all contributions to it.