In
mathematics, especially
general topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometr ...
and
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
, an exhaustion by compact sets of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
is a nested sequence of
compact subset
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
s
of
(i.e.
), such that
is contained in the
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
of
, i.e.
for each
and
. A space admitting an exhaustion by compact sets is called exhaustible by compact sets.
For example, consider
and the sequence of closed balls
.
Occasionally some authors drop the requirement that
is in the interior of
, but then the property becomes the same as the space being
σ-compact, namely a countable union of compact subsets.
Properties
The following are equivalent for a topological space
:
#
is exhaustible by compact sets.
#
is
σ-compact and
weakly locally compact.
#
is
Lindelöf and weakly locally compact.
(where ''weakly locally compact'' means
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 ...
in the weak sense that each point has a compact neighborhood).
The
hemicompact property is intermediate between exhaustible by compact sets and σ-compact. Every space exhaustible by compact sets is hemicompact. Every hemicompact space is σ-compact. The reverse implications do not hold. For example, the
Arens-Fort space and the
Appert space are hemicompact, but not exhaustible by compact sets (because not weakly locally compact). And the set
of rational numbers with the usual topology is σ-compact, but not hemicompact.
Every
regular space
In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' can b ...
exhaustible by compact sets is
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is norma ...
.
Notes
References
*
Leon Ehrenpreis
Eliezer 'Leon' Ehrenpreis (May 22, 1930 – August 16, 2010, Brooklyn) was a mathematician at Temple University who proved the Malgrange–Ehrenpreis theorem, the fundamental theorem about differential operators with constant coefficients. He pre ...
, ''Theory of Distributions for Locally Compact Spaces'',
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, 1982. .
*
Hans Grauert
Hans Grauert (8 February 1930 in Haren, Emsland, Germany – 4 September 2011) was a German mathematician. He is known for major works on several complex variables, complex manifolds and the application of sheaf theory in this area, which i ...
and
Reinhold Remmert
Reinhold Remmert (22 June 1930 – 9 March 2016) was a German mathematician. Born in Osnabrück, Lower Saxony, he studied mathematics, mathematical logic and physics in Münster. He established and developed the theory of complex-analytic space ...
, ''Theory of Stein Spaces'',
Springer Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 i ...
(Classics in Mathematics), 2004. .
*
External links
*
* {{cite web , title=Existence of exhaustion by compact sets , url=https://math.stackexchange.com/questions/1360900 , website=Mathematics Stack Exchange
Compactness (mathematics)
Mathematical analysis
General topology