Compact space
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"Compact" redirects here. For other uses, see
Compact (disambiguation).
In
mathematics, a
subset of
Euclidean space Rn is called compact if
it is
closed and
bounded. For example, in R, the closed
unit interval [0, 1] is compact, but the set of
integers
Z is not (it is not bounded) and neither is the half-open interval
[0, 1) (it is not closed).
A more modern approach is to call a
topological space compact if each of its
open covers has a
finite
subcover. The
Heine�Borel theorem shows that this definition is equivalent to "closed
and bounded" for subsets of Euclidean space. Note: Some authors such as
Bourbaki use the term "quasi-compact" instead, and reserve the
term "compact" for topological spaces that are
Hausdorff and "quasi-compact".
A single compact set is sometimes referred to as a compactum;
following the
Latin
second declension (neuter), the corresponding plural form is compacta.
Definitions
Compactness
of subsets of Rn
For any subset
of
Euclidean space Rn, the following four conditions
are equivalent:
- Every
open cover has a finite
subcover. This is the most commonly used definition.
- Every
sequence in the set has a
convergent subsequence, the limit point of which belongs to the set.
- Every infinite subset of the set has an
accumulation point in the set.
- The set is
closed
and
bounded. This is the condition that is easiest to verify, for example a
closed
interval or closed n-ball.
In other spaces, these conditions may or may not be equivalent, depending on
the properties of the space.
Note that while compactness is a property of the set itself (with its
topology), closedness is relative to a space it is in; above "closed" is used in
the sense of closed in Rn. A set which is closed in
e.g. Qn is typically not closed in Rn,
hence not compact.
Compactness
of topological spaces
The "finite subcover" property from the previous paragraph is more abstract
than the "closed and bounded" one, but it has the distinct advantage that it can
be given using the
subspace topology on a subset of Rn, eliminating
the need of using a metric or an ambient space. Thus, compactness is a
topological property. In a sense, the closed unit interval [0,1] is
intrinsically compact, regardless of how it is embedded in R or Rn.
Other forms of compactnessThere are a number of topological properties which are equivalent to
compactness in
metric spaces, but are inequivalent in general topological spaces. These
include the following.
- Sequentially compact: Every
sequence
has a convergent subsequence.
- Countably compact: Every countable open cover has a finite
subcover. (Or, equivalently, every infinite subset has an ω-accumulation
point.)
-
Pseudocompact : Every real-valued
continuous
function on the space is bounded.
-
Weakly countably compact (or limit point compact): Every infinite
subset has an
accumulation point.
While all these conditions are equivalent for
metric
spaces, in general we have the following implications:
- Compact spaces are countably compact.
- Sequentially compact spaces are countably compact.
- Countably compact spaces are pseudocompact and weakly countably compact.
Not every countably compact space is compact; an example is given by the
first uncountable ordinal with the order topology. Not every compact space is
sequentially compact; an example is given by 2[0,1], with the product
topology.
A metric space is called pre-compact or
totally bounded if any sequence has a Cauchy subsequence; this can be
generalised to
uniform spaces. For complete metric spaces this is equivalent to
compactness. See
relatively compact for the topological version.
Another related notion which (by most definitions) is strictly weaker than
compactness is
compactness.
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