In mathematics, a **Cantor space**, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a **Cantor space** if it is homeomorphic to the Cantor set. In set theory, the topological space 2^{ω} is called "the" Cantor space.

The Cantor set itself is 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 or 2^{ω} (where 2 denotes the 2-element set {0,1} with the discrete topology). A point in 2^{ω} is an infinite binary sequence, that is a sequence which assumes only the values 0 or 1. Given such a sequence *a*_{0}, *a*_{1}, *a*_{2},..., one can map it to the real number

This mapping gives a homeomorphism from 2^{ω} onto the Cantor set, demonstrating that 2^{ω} is indeed a Cantor space.

Cantor spaces occur abundantly 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.

A topological characterization of Cantor spaces is given by Brouwer's theorem:^{ [1] }

The topological property of having a base consisting of clopen sets is sometimes known as "zero-dimensionality". Brouwer's theorem can be restated as:

This theorem is also equivalent (via Stone's representation theorem for Boolean algebras) 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 many properties of Cantor spaces can be established using 2^{ω}, because its construction as a product makes it amenable to analysis.

Cantor spaces have the following properties:

- The cardinality of any Cantor space is , that is, the cardinality of the continuum.
- The product of two (or even any finite or countable number of) Cantor spaces is a Cantor space. Along with the Cantor function, this fact can be used to construct space-filling curves.
- A (non-empty) Hausdorff topological space is compact metrizable if and only if it is a continuous image of a Cantor space.
^{ [2] }^{ [3] }^{ [4] }

Let *C*(*X*) denote the space of all real-valued, bounded continuous functions on a topological space *X*. Let *K* denote a compact metric space, and Δ denote the Cantor set. Then the Cantor set has the following property:

*C*(*K*) is isometric to a closed subspace of*C*(Δ).^{ [5] }

In general, this isometry is not unique, and thus is not properly a universal property in the categorical sense.

- The group of all homeomorphisms of the Cantor space is simple.
^{ [6] }

In mathematics, more specifically in functional analysis, a **Banach space** is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, more specifically in general topology, **compactness** is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

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In mathematics, the **real line**, or **real number line** is the line whose points are the real numbers. That is, the real line is the set **R** of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space, a metric space, a topological space, a measure space, or a linear continuum.

The **Baire category theorem** (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.

In topology, a **discrete space** is a particularly simple example of a topological space or similar structure, one in which the points form a *discontinuous sequence*, meaning they are *isolated* from each other in a certain sense. The discrete topology is the finest topology that can be given on a set, i.e., it defines all subsets as open sets. In particular, each singleton is an open set in the discrete topology.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In mathematics, a **Baire space** is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-Louis Baire who introduced the concept.

In mathematics, the **Hilbert cube**, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube.

In topology and related branches of mathematics, a **totally disconnected space** is a topological space that is maximally disconnected, in the sense that it has no non-trivial connected subsets. In every topological space, the singletons are connected; in a totally disconnected space, these are the *only* connected subsets.

In mathematical analysis, a **space-filling curve** is a curve whose range contains the entire 2-dimensional unit square. Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called *Peano curves*, but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano.

In the mathematical discipline of general topology, a **Polish space** is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.

In topology and related areas of mathematics, a **topological property** or **topological invariant** is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space *X* possesses that property every space homeomorphic to *X* possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In topology, a branch of mathematics, a **topological manifold** is a topological space which locally resembles real *n*-dimensional space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. All manifolds are topological manifolds by definition, but many manifolds may be equipped with additional structure. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" any additional structure the manifold has.

In topology, a branch of mathematics, a **retraction** is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a **retract** of the original space. A **deformation retraction** is a mapping that captures the idea of *continuously shrinking* a space into a subspace.

In mathematics, a **polyadic space** is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete topological space.

- ↑ Brouwer, L. E. J. (1910), "On the structure of perfect sets of points" (PDF),
*Proc. Koninklijke Akademie van Wetenschappen*,**12**: 785–794. - ↑ N.L. Carothers,
*A Short Course on Banach Space Theory*, London Mathematical Society Student Texts**64**, (2005) Cambridge University Press.*See Chapter 12* - ↑ Willard,
*op.cit.*,*See section 30.7* - ↑ https://imgur.com/a/UDgthQm
- ↑ Carothers,
*op.cit.* - ↑ R.D. Anderson,
*The Algebraic Simplicity of Certain Groups of Homeomorphisms*, American Journal of Mathematics**80**(1958), pp. 955-963.

- Kechris, A. (1995).
*Classical Descriptive Set Theory*(Graduate Texts in Mathematics 156 ed.). Springer. ISBN 0-387-94374-9.

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