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
Condensed sets are basic objects in condensed mathematics, whose aim is to provide a convenient setting in the framework for working with algebraic objects that are equipped with sort of a topology. A related alternative is provided by pyknotic sets.
A condensed set is a sheaf of sets on the pro-étale site of a point — in other words, on the category of profinite spaces with finite jointly surjective families of maps as covers — which is the colimit of a small diagram of representables (a small sheaf?).
That is, a condensed set is a functor
such that the natural maps
and
are bijections for any profinite sets $S$ and $S'$, whereas the natural fork
is an equalizer for any surjection of profinite sets $S'\to S$.
Scholze, p.7 modifies this definition to deal with size issues:
For any uncountable strong limit cardinal $\kappa$, the category of $\kappa$-condensed sets is the category of sheaves on the site of profinite sets of cardinality less than $\kappa$, with finite jointly surjective families of maps as covers.
The category of condensed sets is then the (large) colimit of the category of $\kappa$-condensed sets along the filtered poset of all uncountable strong limit cardinals $\kappa$, hence is the category of small sheaves?.
Condensed sets form a pretopos.
See \cite[Proposition 1.7]{ScholzeLCM} for the following proposition.
The forgetful functor from the category of topological spaces to condensed sets is a faithful functor. It becomes fully faithful when restricted to compactly generated spaces. (In the case of κ-condensed sets, one must take κ-compactly generated spaces instead.)
This functor admits a left adjoint, which sends a condensed set $T$ to the topological space given by the underlying set $T(*)$ of $T$ equipped with the quotient topology induced by the map
where $S$ runs over all (κ-small) profinite sets mapping into $T$. The counit of this adjunction coincides with the counit $X^{cg}\to X$ of the adjunction between (κ-small) compactly generated spaces and topological spaces.
The left adjoint exists also for topological groups and other algebraic structures, but in this case, the underlying set is not $T(*)$.
Last revised on April 10, 2021 at 14:48:46. See the history of this page for a list of all contributions to it.