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 ...
, 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'' is sequentially compact if every
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of points in ''X'' has a
convergent subsequence converging to a point in
.
Every
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
is naturally a topological space, and for metric spaces, the notions of
compactness
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ...
and sequential compactness are equivalent (if one assumes
countable choice). However, there exist sequentially compact topological spaces that are not compact, and compact topological spaces that are not sequentially compact.
Examples and properties
The space of all
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s with the
standard topology is not sequentially compact; the sequence
given by
for all
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s ''
'' is a sequence that has no convergent subsequence.
If a space is a
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
, then it is sequentially compact if and only if it is
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 ...
. The
first uncountable ordinal with the
order topology is an example of a sequentially compact topological space that is not compact. The
topological product of
copies of the
closed unit interval is an example of a compact space that is not sequentially compact.
Related notions
A topological space ''
'' is said to be
limit point compact if every infinite subset of ''
'' has a
limit point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x contains a point of S other than x itself. A ...
in ''
'', and
countably compact if every countable
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family 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\su ...
has a finite subcover. In a
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
, the notions of sequential compactness, limit point compactness, countable compactness and
compactness
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ...
are all equivalent (if one assumes the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
).
In a
sequential (Hausdorff) space sequential compactness is equivalent to countable compactness.
There is also a notion of a one-point sequential compactification—the idea is that the non convergent sequences should all converge to the extra point.
[Brown, Ronald, "Sequentially proper maps and a sequential
compactification", J. London Math Soc. (2) 7 (1973)
515-522.
]
See also
*
*
*
*
Notes
References
*
*
Steen, Lynn A. and
Seebach, J. Arthur Jr.; ''
Counterexamples in Topology'', Holt, Rinehart and Winston (1970). .
*
Compactness (mathematics)
Properties of topological spaces
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