A metrizable
compact space.
Examples of compacta are: a segment, a circle, an
-dimensional
cube, ball or sphere, the
Cantor set,
the
Hilbert cube;
an
-dimensional
Euclidean space is not a compactum, but a subset of it is a compactum
if and only if it is closed and bounded. A closed subset of
a compactum is a compactum and every compactum is homeomorphic to
a closed subset of the Hilbert cube
(Urysohn's theorem).
In order that there exist a homeomorphism of a compactum into a
Euclidean space, it is necessary and sufficient
that the compactum be finite-dimensional (the
Pontryagin–Nöbeling theorem).
A continuous image of a compactum that is a
-space
is a compactum, and every compactum is the continuous
image of the Cantor set
(Aleksandrov's theorem).
The product of a
finite or countable set of compacta is a compactum. Every compactum
is separable; among the Hausdorff compact spaces, the compacta are
characterized by the property that they possess a finite or countable
basis. A compactum is also characterized by the fact that it
is totally bounded with respect to any metric
compatible with its topology
(Hausdorff's theorem).
The compacta form one of the most important classes
of topological spaces. The property that a metrizable space
be a compactum is equivalent to each of the following properties.
1)
From any countable open covering of the space
one can select a finite subcovering (an analogue of the
Heine–Borel–Lebesgue covering theorem
on covering a line segment by intervals; cf. also
Borel–Lebesgue covering theorem).
2)
Any countable system of non-empty closed subsets
of
such that
,
has a non-empty intersection (a generalization of
Cantor's principle of nested intervals).
3)
Any sequence of points in
has a convergent subsequence in
(a generalization of the
Bolzano–Weierstrass theorem).
4)
Any infinite subset of
has at least one limit point in
(a generalization of the
Bolzano–Weierstrass theorem).
5)
Any continuous function on
is bounded (a generalization of
Weierstrass' theorem).
6)
Any continuous function on
attains its maximum (minimum) value at some
point (a generalization of Weierstrass' theorem).
7)
is totally bounded and complete with respect to any metric compatible with its topology.
Any continuous function on a compactum
is uniformly continuous with respect to any metric compatible with the topology of
(a generalization of the
Heine–Cantor theorem).