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
Not to be confused with the notion of coherence space in models of linear logic.
A coherent space (alias spectral topological space) is a topological space which is homeomorphic to the spectrum of a commutative ring, hence to the topological space underlying an affine scheme.
Equivalently, it is a compact sober topological space whose collection of compact open subsets is closed under finite intersections and forms a topological base.
Morphisms of coherent spaces are continuous maps such that preimages of compact open subsets are again compact.
Passing from a coherent space to its lattice of compact open subsets establishes a contravariant equivalence from the category of coherent spaces to the category of (bounded) distributive lattices.
A coherent space is Hausdorff if and only if it is a Stone space. Under Stone duality for coherent spaces, this corresponds to the fact that in a distributive lattice $L$ every element has a complement if and only if $L$ is a Boolean algebra.
In particular, restricting the Stone duality equivalence between coherent spaces and distributive lattices to Stone spaces and Boolean algebras recovers the classical Stone duality.
A locale is coherent if its compact elements are closed under finite meets and any element is a join of compact elements.
A morphism of coherent locales is a morphism $f$ of locales such that $f^*$ preserves compact elements.
Assuming the axiom of choice, coherent locales are automatically spatial.
The category of coherent locales is contravariantly equivalent to the category of distributive lattices. This statement does not depend on the axiom of choice, unlike the point-set version above.
Michael Barr, John Kennison, Robert Raphael, Countable meets in coherent spaces with applications to the cyclic spectrum, Theory Appl. Categories 25 19 (2011), 508–232 [tac:25-19]
Max Dickmann, Niels Schwartz, Marcus Tressl, Spectral Spaces. New Mathematical Monographs 35 (2019). Cambridge: Cambridge University Press. ISBN 9781107146723.
See also:
Last revised on October 25, 2023 at 06:39:20. See the history of this page for a list of all contributions to it.