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
In general, the concept of sheaf is relative to a choice of site, hence to a choice of small category $\mathcal{C}$ equipped with a coverage. Given this, the category of sheaves on $\mathcal{C}$ is the full subcategory of the category of presheaves $[\mathcal{C}^{op}, Set]$ on those presheaves which satisfy the sheaf condition:
Often sheaves are introduced and discussed in the more restrictive sense of sheaves on a topological space. This is indeed a special case: A sheaf on a topological space $X$ is a sheaf on the site $Op(X)$ whose underlying category is the category of open subsets of $X$, and whose coverage consists of the open covers of topological spaces. It is usual to write $Sh(X)$ as shorthand for the resulting category of sheaves, but the more systematic name is “$Sh(Op(X))$”:
A topos which is equivalent to a category of sheaves on a topological spaces, this way, is called a spatial topos.
Similarly, if $X$ is a locale, there is its frame of opens $Op(X)$ and one obtains the sheaf topos $Sh(X) \coloneqq Sh(Op(X))$ as above. A topos in the essential image of this construction is called a localic topos. A sober topological space is canonically identified as a locale, and hence the category of sheaves over a sober topological space is a localic topos.
Restricted to sober topological spaces, the construction of forming categories of sheaves is a fully faithful functor to the category Topos of toposes with geometric morphisms between them:
(MacLaneMoerdijk, theorem IX.3 1) or as (Johnstone, lemma C.1.2.2)
For more see at locale the section Localic reflection.
Examples of spatial toposes which are not manifestly spatial, by usual their definition, include the following:
The Bohr topos of a C*-algebra
The category of hypergraphs
The Sierpinski topos
Last revised on February 21, 2024 at 15:35:08. See the history of this page for a list of all contributions to it.