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. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... This word should not be confused with homomorphism. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...

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 {0, 1}. This is usually written as 2^N or 2^ω (where 2 denotes the 2-element set {0,1} with the discrete topology). A point in 2^N 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 In mathematics the term countable set is used to describe the size of a set, e. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...

It is not difficult to see that this mapping is a homeomorphism from 2^N onto the Cantor set, and hence that 2^N is indeed a Cantor space. The term "Cantor space" can more generally be used to refer to any infinite product of the discrete 2-point space, whether countable or uncountable. The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...

A topological characterization of Cantor spaces is given by Brouwer's theorem: Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis. ...

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: In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology, a point x of a set S is called an isolated point, if there exists a neighborhood of x not containing other points of S. In particular, in an Euclidean space (or in a metric space), x is an isolated point of S, if one can find an... In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases... In topology, a clopen set (or closed-open set) in a topological space is a set which is both open and closed. ...

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 representation theorem for Boolean algebras) to the fact that any two countable atomless Boolean algebras are isomorphic. In mathematics a derived set is a construction in point-set topology that consists of taking the set of limit points of a given subset S of a topological space X. The derived set of S is usually denoted by S′. A subset S of a topological space X is... In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... A metrizable space is a topological space that is homeomorphic to a metric space. ... In mathematics, Stones representation theorem for Boolean algebras, named in honor of Marshall H. Stone, is the duality between the category of Boolean algebras and the category of Stone spaces, i. ... Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ...

As can be expected from Brouwer's theorem, Cantor spaces appear in several forms. But it is usually easiest to deal with 2^N, 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. In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ...

Using this last fact and the Cantor function, it is easy to construct space-filling curves. In mathematics, the Cantor function is a function c : [0,1] → [0,1] defined as follows: Express x in base 3. ... Space-filling curves or Peano curves are curves, first described by Giuseppe Peano, whose ranges contain the entire 2-dimensional unit square (or the 3-dimensional unit cube). ...

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. Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...

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.