In the mathematical field of point-set topology, a **continuum** (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. **Continuum theory** is the branch of topology devoted to the study of continua.

- A continuum that contains more than one point is called
**nondegenerate**. - A subset
*A*of a continuum*X*such that*A*itself is a continuum is called a**subcontinuum**of*X*. A space homeomorphic to a subcontinuum of the Euclidean plane**R**^{2}is called a**planar continuum**. - A continuum
*X*is**homogeneous**if for every two points*x*and*y*in*X*, there exists a homeomorphism*h*:*X*→*X*such that*h*(*x*) =*y*. - A
**Peano continuum**is a continuum that is locally connected at each point. - An indecomposable continuum is a continuum that cannot be represented as the union of two proper subcontinua. A continuum
*X*is**hereditarily indecomposable**if every subcontinuum of*X*is indecomposable. - The
**dimension**of a continuum usually means its topological dimension. A one-dimensional continuum is often called a**curve**.

- An
**arc**is a space homeomorphic to the closed interval [0,1]. If*h*: [0,1] →*X*is a homeomorphism and*h*(0) =*p*and*h*(1) =*q*then*p*and*q*are called the**endpoints**of*X*; one also says that*X*is an arc from*p*to*q*. An arc is the simplest and most familiar type of a continuum. It is one-dimensional, arcwise connected, and locally connected. - The topologist's sine curve is a subset of the plane that is the union of the graph of the function
*f*(*x*) = sin(1/*x*), 0 <*x*≤ 1 with the segment −1 ≤*y*≤ 1 of the*y*-axis. It is a one-dimensional continuum that is not arcwise connected, and it is locally disconnected at the points along the*y*-axis. - The Warsaw circle is obtained by "closing up" the topologist's sine curve by an arc connecting (0,−1) and (1,sin(1)). It is a one-dimensional continuum whose homotopy groups are all trivial, but it is not a contractible space.

- An
is a space homeomorphic to the closed ball in the Euclidean space*n*-cell**R**^{n}. It is contractible and is the simplest example of an*n*-dimensional continuum. - An
is a space homeomorphic to the standard n-sphere in the (*n*-sphere*n*+ 1)-dimensional Euclidean space. It is an*n*-dimensional homogeneous continuum that is not contractible, and therefore different from an*n*-cell. - The Hilbert cube is an infinite-dimensional continuum.
- Solenoids are among the simplest examples of indecomposable homogeneous continua. They are neither arcwise connected nor locally connected.
- The Sierpinski carpet, also known as the
*Sierpinski universal curve*, is a one-dimensional planar Peano continuum that contains a homeomorphic image of any one-dimensional planar continuum. - The pseudo-arc is a homogeneous hereditarily indecomposable planar continuum.

There are two fundamental techniques for constructing continua, by means of *nested intersections* and *inverse limits*.

- If {
*X*_{n}} is a nested family of continua, i.e.*X*_{n}⊇*X*_{n+1}, then their intersection is a continuum.

- If {

- If {(
*X*_{n},*f*_{n})} is an inverse sequence of continua*X*_{n}, called the**coordinate spaces**, together with continuous maps*f*_{n}:*X*_{n+1}→*X*_{n}, called the**bonding maps**, then its inverse limit is a continuum.

- If {(

A finite or countable product of continua is a continuum.

In topology and related branches of mathematics, a **connected space** is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces.

In mathematics, a **diffeomorphism** is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.

In the mathematical field of topology, a **homeomorphism**, **topological isomorphism**, or **bicontinuous function** is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called **homeomorphic**, and from a topological viewpoint they are the same. The word *homeomorphism* comes from the Greek words *ὅμοιος* (*homoios*) = similar or same and *μορφή* (*morphē*) = shape, form, introduced to mathematics by Henri Poincaré in 1895.

In the part of mathematics referred to as topology, a **surface** is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.

In mathematics, **topology** is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.

In topology and related branches of mathematics, a **topological space** may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

In topology and related branches of mathematics, a topological space is called **locally compact** if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

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.

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 the branch of mathematics known as topology, the **topologist's sine curve** or **Warsaw sine curve** is a topological space with several interesting properties that make it an important textbook example.

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 mathematics, **low-dimensional topology** is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

In mathematics, a topological space *X* is **contractible** if the identity map on *X* is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension n.

In topology, a branch of mathematics, a **topological manifold** is a topological space which locally resembles real *n*-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.

In mathematics, a **solenoid** is a compact connected topological space that may be obtained as the inverse limit of an inverse system of topological groups and continuous homomorphisms

In point-set topology, an **indecomposable continuum** is a continuum that is indecomposable, i.e. that cannot be expressed as the union of any two of its proper subcontinua. In 1910, L. E. J. Brouwer was the first to describe an indecomposable continuum.

In general topology, the **pseudo-arc** is the simplest nondegenerate hereditarily indecomposable continuum. The pseudo-arc is an arc-like homogeneous continuum, and played a central role in the classification of homogeneous planar continua. R.H. Bing proved that, in a certain well-defined sense, most continua in **R**^{n}, *n* ≥ 2, are homeomorphic to the pseudo-arc.

In topology and other branches of mathematics, a topological space *X* is **locally connected** if every point admits a neighbourhood basis consisting entirely of open, connected sets.

- Sam B. Nadler, Jr,
*Continuum theory. An introduction*. Pure and Applied Mathematics, Marcel Dekker. ISBN 0-8247-8659-9.

- Open problems in continuum theory
- Examples in continuum theory
- Continuum Theory and Topological Dynamics, M. Barge and J. Kennedy, in Open Problems in Topology, J. van Mill and G.M. Reed (Editors) Elsevier Science Publishers B.V. (North-Holland), 1990.
- Hyperspacewiki

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.