The subset of the real interval
consisting of all numbers of the form
,
where
is 0 or 2. It is geometrically described as follows (see
Fig.): One removes from
its middle third
;
one then removes from the remaining intervals
,
their middle thirds
and
;
then the middle thirds of the four remaining intervals, etc. Then
what remains after removal of all these intervals
(adjacent intervals),
the total of whose length is 1, is the
Cantor perfect set
(Cantor set;
Cantor ternary set;
Cantor discontinuum).
It is nowhere dense in the real line but has the cardinality of the continuum.
Figure: c020250a
From a topological point of view, the Cantor set
is a zero-dimensional, perfect, metrizable compactum (that is, without isolated
points); such a compactum is unique up to a homeomorphism.
All bounded, perfect, nowhere-dense subsets of the real line are similar
sets. The Cantor set is homeomorphic to a countable product
of copies of a two-point space
,
and is the space of the topological group
.
The Cantor set is universal in two senses: 1)
first of all, every zero-dimensional regular Hausdorff space with a countable
base is homeomorphic to a subset of the Cantor set; 2)
secondly, every metrizable compactum is a continuous image
of the Cantor set
(Aleksandrov's theorem).
This theorem marks the
beginning of the theory of dyadic compacta and shows that many
compacta are similar to one another from the functional point
of view. In particular, all perfect compacta have the same
Boolean algebra of canonical open sets. The existence of special mappings
from the Cantor set onto compacta allows one to prove that
the Banach algebras of all continuous functions on two arbitrary perfect
metrizable compacta (for example, on an interval and on a
square) are linearly homeomorphic. Furthermore, the Cantor set and the
possibility of mapping it onto an arbitrary metrizable compactum lies
at the basis of the construction of many interesting examples in
topology and function theory. One of them is the so-called
Cantor staircase,
which is the graph of a continuous monotone mapping of
onto itself, the derivative of which is defined and equal
to zero on an open set of measure 1. Although
the standard Cantor set has measure zero, there exists
nowhere-dense perfect compacta on the unit interval with measure arbitrarily close to 1.