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
A Cauchy space is a generalisation of a metric space with a bare minimum of structure for the concepts of Cauchy sequence, Cauchy-continuous map, and Cauchy completion to make sense. Topologically (that is, up to continuous maps), any Cauchy space is a convergence space, but not much more than that. Besides Cauchy sequences, we can also speak more generally of Cauchy nets and Cauchy filters in a Cauchy space; in fact, the usual definition is in terms of Cauchy filters.
A Cauchy space is a set $S$ together with a collection of proper filters declared to be Cauchy filters. These must satisfy axioms:
That is, the set of Cauchy filters is a local filter of proper filters that contains all principal ultrafilters (sort of a tongue twister).
The definition can also be phrased in terms of nets; a Cauchy net is a net whose eventuality filter is Cauchy. In particular, a Cauchy sequence is a sequence whose eventuality filter is Cauchy.
The morphisms of Cauchy spaces are the Cauchy-continuous functions; a function $f$ between Cauchy spaces is Cauchy-continuous if $f(F)$ is a (base of a) Cauchy filter whenever $F$ is. In this way, Cauchy spaces form a concrete category $Cau$.
Any metric space is a Cauchy space: $F$ is a Cauchy filter iff it has elements of arbitrarily small diameter. This reconstructs the usual definitions of Cauchy sequence and Cauchy-continuous map for metric spaces. (In particular, a map between metric spaces is Cauchy-continuous iff it maps every Cauchy sequence to a Cauchy sequence; the result for general nets follows since a metric space is sequential.) The forgetful functor from $Met$ (metric spaces and short maps) to $Cau$ is faithful but not full.
More generally, any uniform space is a Cauchy space: $F$ is a Cauchy filter if, given any entourage $U$, $A \times A \subseteq U$ for some $A \in F$. This reconstructs the usual definitions of Cauchy net and Cauchy-continuous map for uniform spaces. (In general, we need nets rather than just sequences here.) A map between uniform spaces is Cauchy continuous iff its restrictions to precompact subspaces are always uniformly continuous. The forgetful functor from $Unif$ (uniform spaces and uniformly continuous maps) to $Cau$ is faithful but still not full.
Assuming the ultrafilter principle and excluded middle, we may take any collection $U$ of free ultrafilters and define a proper filter $F$ to be Cauchy if (hence iff) every free ultrafilter that refines $F$ belongs to $U$.
Every Cauchy space is a convergence space; $F \to x$ if the intersection of $F$ with the principal ultrafilter $F_x$ is Cauchy. Note that any convergent proper filter must be Cauchy. Conversely, if every Cauchy filter is convergent, then the Cauchy space is called complete.
The set of Cauchy filters on a Cauchy space has a natural Cauchy structure which is complete and (as a convergence space) preregular; we identify the indistinguishable Cauchy filters to get a Hausdorff space, the Hausdorff completion of the original Cauchy space. The complete Hausdorff Cauchy spaces thus form a reflective subcategory of $Cau$. This completion agrees with the completion of a metric or uniform space; that is, Cauchy completion, even of a metric space, is an operation on its Cauchy structure only.
Conversely, every Hausdorff convergence space becomes a complete Hausdorff Cauchy space upon declaring that the Cauchy filters are precisely the convergent proper filters. The convergence structure on this Cauchy space matches the original convergence structure.
A Cauchy space $S$ is precompact (or totally bounded) if every proper filter is contained in a Cauchy filter. Equivalently (assuming the ultrafilter principle), $S$ is precompact iff every ultrafilter is Cauchy. A Cauchy space is compact (as a convergence space) iff it is both complete and precompact. Conversely, it is precompact iff its completion is compact.
Last revised on May 4, 2022 at 03:08:52. See the history of this page for a list of all contributions to it.