In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Stone–Čech remainder of a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''X'', also called the corona or corona set, is the
complement
Complement may refer to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class collections into complementary sets
* Complementary color, in the visu ...
of the space in its
Stone–Čech compactification
In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a Universal property, universal map from a topological space ''X'' to a Compact space, compact Ha ...
β''X''.
A topological space is said to be
σ-compact if it is the union of
countably
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
many
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
subspaces, and
locally compact
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
if every point has a
neighbourhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...
with compact
closure. The Stone–Čech remainder of a σ-compact and locally compact
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
is a
sub-Stonean space
In topology, a sub-Stonean space is a locally compact Hausdorff space such that any two open σ-compact disjoint subsets have disjoint compact closures. Related, an F-space, introduced by , is a completely regular Hausdorff space for which every f ...
, i.e., any two
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979
* ''Open'' (Go ...
σ-compact
disjoint subsets have disjoint compact closures.
See also
*
Corona theorem In mathematics, the corona theorem is a result about the spectrum of the bounded holomorphic functions on the open unit disc, conjectured by and proved by .
The commutative Banach algebra and Hardy space ''H''∞ consists of the bounded ho ...
*
Corona algebra In mathematics, the multiplier algebra, denoted by ''M''(''A''), of a C*-algebra ''A'' is a unital C*-algebra that is the largest unital C*-algebra that contains ''A'' as an ideal in a "non-degenerate" way. It is the noncommutative generalization of ...
, a non-commutative analogue of the corona set.
References
*{{Citation , last1=Grove , first1=Karsten , last2=Pedersen , first2=Gert Kjærgård , title=Sub-Stonean spaces and corona sets , doi=10.1016/0022-1236(84)90028-4 , mr=735707 , year=1984 , journal=
Journal of Functional Analysis , issn=0022-1236 , volume=56 , issue=1 , pages=124–143, doi-access=free
Topology