Cantor space
In mathematics, the term Cantor space is sometimes used to denote
the topological abstraction of the classical Cantor set:
A topological space is a
Cantor space if it is homeomorphic to the Cantor set.
The Cantor set itself is of course a Cantor space. But
the canonical example of a Cantor space is the countably infinite topological product of the discrete 2-point space
. This is usually written as 2N or by 2ω
(where 2 denotes the 2-element set
with the discrete topology).
A point in 2N is an
infinite binary sequence, that is a sequence which
assumes only the values 0 or 1. Given such a
sequence a1, a2,a3,...
one can map it to the real number
It is not difficult to see that this mapping is a
homeomorphism from 2N
onto the Cantor set, and hence that
2N is indeed a Cantor space.
A topological characterization of Cantor spaces is given
by Brouwer's theorem:
- Any two non-empty compact Hausdorff spaces without isolated points and having countable bases consisting of clopen sets are homeomorphic to each other.
(The topological property of having a base consisting
of clopen sets is sometimes known as "zero-dimensionality".)
This theorem can be restated as:
- A topological space is a Cantor space if and only if it is non-empty, perfect, compact, totally disconnected, and metrizable.
It is also equivalent (via Stone's duality)
to the fact that any two countable atomless
Boolean algebras are isomorphic.
As can be expected from Brouwer's theorem, Cantor spaces
appear in several forms. But it is usually easiest to deal with
2N, since because of
its special product form, many topological and other
properties are brought out very explicitly.
For example, it becomes obvious that the cardinality of
any Cantor space is  , that is,
the cardinality of the continuum. Also clear is the
fact that the product of two
(or even any finite or countable number of) Cantor spaces
is a Cantor space - an important fact about Cantor spaces.
Using this last fact and the Cantor function, it is easy
to construct space-filling curves.
Cantor spaces occur in abundance in real analysis.
For example they exist as subspaces in every perfect,
complete metric space. (To see this, note that in
such a space, any non-empty perfect set contains
two disjoint non-empty perfect subsets of arbitrarily
small diameter, and so one can imitate the construction
of the usual Cantor set.) Also, every uncountable,
separable, completely metrizable space contains
Cantor spaces as subspaces. This includes most of
the common type of spaces in real analysis.
As a corollary, we see that separable, completely
metrizable spaces satisfy the Continuum hypothesis:
Every such space is either countable or has the
cardinality of the continuum.
Compact metric spaces are also closely related to
Cantor spaces: A Hausdorff topological space is compact
metrizable if and only if it is a continuous image
of a Cantor space.
Referenced By
Baire space | CantorSet | Cantor set | First category | List of general topology topics | List of mathematical topics | List of mathematical topics (A-C) | List of mathematics topics | List of real analysis topics | Meager set | Meagre set | Peano's curves | Peano curve | Second category | Space-filling curve
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