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
symmetric monoidal (∞,1)-category of spectra
Pyknotic sets (from the ancient Greek πυκνός, meaning thick, dense, or compact) aim to provide a convenient setting in the framework of homotopical algebra for working with algebraic objects that have some sort of topology on them.
A pyknotic set is a sheaf of sets on the site of compact Hausdorff topological spaces whose underlying set has rank smaller than the smallest strongly inaccessible cardinal. The Grothendieck topology on the latter category is generated by finite jointly effective epimorphic families of maps.
Pyknotic sets form a coherent topos, whereas the closely related category of condensed sets does not form a topos (BarHai 19, Sec. 0.3).
The global sections functor is given by evaluation at the one-point compactum. This has a left adjoint sending a set, $X$, to the discrete pyknotic set attached to $X$, and a right adjoint to the indiscrete pyknotic set attached to $X$. The left adjoint, the discrete functor, does not preserve limits and so does not possess a further left adjoint. The topos of pyknotic sets is thus not cohesive (BarHai 19, 2.2.14).
Compactly generated topological spaces embed fully faithfully into pyknotic sets, in a manner that preserves limits.
One key point, however, is that the relationship between compactly generated topological spaces and pyknotic sets is dual to the relationship between compactly generated topological spaces and general topological spaces: in topological spaces, compactly generated topological spaces are stable under colimits but not limits; in pyknotic sets,compactly generated topological spaces are stable under limits but not colimits (BarHai 19, 0.2.3).
This construction generalizes to pyknotic objects for any (∞,1)-category.
For example locally compact abelian groups, normed rings, and complete locally convex topological vector spaces are pyknotic objects.
quasi-topological space (roughly, the concrete pyknotic sets)
Last revised on June 11, 2019 at 15:37:22. See the history of this page for a list of all contributions to it.