In
mathematics, 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 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 subspaces.
A space is said to be σ-locally compact if it is both σ-compact and
locally compact.
Properties and examples
* Every
compact space
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", ...
is σ-compact, and every σ-compact space is
Lindelöf (i.e. every
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alph ...
has a countable
subcover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
). The reverse implications do not hold, for example, standard
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
(R
''n'') is σ-compact but not compact, and the
lower limit topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of inte ...
on the real line is Lindelöf but not σ-compact. In fact, the
countable complement topology The cocountable topology or countable complement topology on any set ''X'' consists of the empty set and all cocountable subsets of ''X'', that is all sets whose Complement (set theory), complement in ''X'' is countable set, countable. It follows th ...
on any uncountable set is Lindelöf but neither σ-compact nor locally compact. However, it is true that any locally compact Lindelöf space is σ-compact.
*A
Hausdorff,
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
that is also σ-compact, must be
locally compact at at least one point.
* If ''G'' is a
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
and ''G'' is locally compact at one point, then ''G'' is locally compact everywhere. Therefore, the previous property tells us that if ''G'' is a σ-compact, Hausdorff topological group that is also a Baire space, then ''G'' is locally compact. This shows that for Hausdorff topological groups that are also Baire spaces, σ-compactness implies local compactness.
* The previous property implies for instance that R
ω is not σ-compact: if it were σ-compact, it would necessarily be locally compact since R
ω is a topological group that is also a Baire space.
* Every
hemicompact space is σ-compact. The converse, however, is not true; for example, the space of
rationals
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
, with the usual topology, is σ-compact but not hemicompact.
* The
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of a finite number of σ-compact spaces is σ-compact. However the product of an infinite number of σ-compact spaces may fail to be σ-compact.
* A σ-compact space ''X'' is second category (respectively Baire) if and only if the set of points at which is ''X'' is locally compact is nonempty (respectively dense) in ''X''.
[Willard, p. 188.]
See also
*
*
*
Notes
References
*
Steen, Lynn A. and
Seebach, J. Arthur Jr.; ''
Counterexamples in Topology
''Counterexamples in Topology'' (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.
In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) h ...
'', Holt, Rinehart and Winston (1970). .
*
{{DEFAULTSORT:Compact Space
Compactness (mathematics)
General topology
Properties of topological spaces