In the mathematical field of point-set topology, a continuum (pl continua) is a nonempty compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 connected

Connected space

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 nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

 metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

, or less frequently, a compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 connected

Connected space

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 nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

 Hausdorff topological space

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

. Continuum theory is the branch of topology devoted to the study of continua

Definitions

• A continuum that contains more than one point is called nondegenerate.

• A subset A of a continuum X such that A itself a continuum is called a subcontinuum of X. A space homeomorphic to a subcontinuum of the Euclidean plane R² 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
  Indecomposable continuum
  In point-set topology, an indecomposable continuum is a continuum that is not the union of any two of its proper subcontinua. The pseudo-arc is an example of a hereditarily indecomposable continuum. L. E. J...
  
   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.

Examples

• 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.

• Topologist's sine curve
  Topologist's sine curve
  In the branch of mathematics known as topology, the topologist's sine curve is a topological space with several interesting properties that make it an important textbook example....
  
   is a subset of the plane which 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
  Topologist's sine curve
  In the branch of mathematics known as topology, the topologist's sine curve is a topological space with several interesting properties that make it an important textbook example....
  
   by an arc connecting (0,−1) and (1,sin(1)). It is a one-dimensional continuum whose homotopy group
  Homotopy group
  In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
  
  s are all trivial, but it is not a contractible space
  Contractible space
  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....
  
  .

• An n-cell is a space homeomorphic to the closed ball in the Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
   Rⁿ. It is contractible and is the simplest example of an n-dimensional continuum.

• An n-sphere is a space homeomorphic to the standard n-sphere in the (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
  Hilbert cube
  In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology...
  
   is an infinite-dimensional continuum.

• Solenoid
  Solenoid (mathematics)
  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...
  
  s are among the simplest examples of indecomposable homogeneous continua. They are neither arcwise connected nor locally connected.

• Sierpinski carpet
  Sierpinski carpet
  The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions . Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional...
  
  , 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.

• Pseudo-arc
  Pseudo-arc
  In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. Pseudo-arc is an arc-like homogeneous continuum. R.H...
  
   is a homogeneous hereditarily indecomposable planar continuum.

Properties

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 {(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
  Inverse limit
  In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
  
   is a continuum.

A finite or countable product of continua is a continuum.

See also

• Linear continuum
  Linear continuum
  In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line.Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two members there is another, and which "lacks gaps" in the...
  
  
• Menger sponge
  Menger sponge
  In mathematics, the Menger sponge is a fractal curve. It is a universal curve, in that it has topological dimension one, and any other curve is homeomorphic to some subset of it. It is sometimes called the Menger-Sierpinski sponge or the Sierpinski sponge...
  
  
• Shape theory (mathematics)
  Shape theory (mathematics)
  Shape theory is a branch of the mathematical field of topology. Homotopy theory is not appropriate for spaces with bad local properties, hence the need for replacement of homotopy theory by a more sophisticated approach...
  
  

In the mathematical field of point-set topology, a continuum (pl continua) is a nonempty compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 connected

Connected space

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 nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

 metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

, or less frequently, a compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 connected

Connected space

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 nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

 Hausdorff topological space

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

. Continuum theory is the branch of topology devoted to the study of continua

Definitions

• A continuum that contains more than one point is called nondegenerate.

• A subset A of a continuum X such that A itself a continuum is called a subcontinuum of X. A space homeomorphic to a subcontinuum of the Euclidean plane R² 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
  Indecomposable continuum
  In point-set topology, an indecomposable continuum is a continuum that is not the union of any two of its proper subcontinua. The pseudo-arc is an example of a hereditarily indecomposable continuum. L. E. J...
  
   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.

Examples

• 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.

• Topologist's sine curve
  Topologist's sine curve
  In the branch of mathematics known as topology, the topologist's sine curve is a topological space with several interesting properties that make it an important textbook example....
  
   is a subset of the plane which 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
  Topologist's sine curve
  In the branch of mathematics known as topology, the topologist's sine curve is a topological space with several interesting properties that make it an important textbook example....
  
   by an arc connecting (0,−1) and (1,sin(1)). It is a one-dimensional continuum whose homotopy group
  Homotopy group
  In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
  
  s are all trivial, but it is not a contractible space
  Contractible space
  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....
  
  .

• An n-cell is a space homeomorphic to the closed ball in the Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
   Rⁿ. It is contractible and is the simplest example of an n-dimensional continuum.

• An n-sphere is a space homeomorphic to the standard n-sphere in the (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
  Hilbert cube
  In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology...
  
   is an infinite-dimensional continuum.

• Solenoid
  Solenoid (mathematics)
  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...
  
  s are among the simplest examples of indecomposable homogeneous continua. They are neither arcwise connected nor locally connected.

• Sierpinski carpet
  Sierpinski carpet
  The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions . Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional...
  
  , 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.

• Pseudo-arc
  Pseudo-arc
  In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. Pseudo-arc is an arc-like homogeneous continuum. R.H...
  
   is a homogeneous hereditarily indecomposable planar continuum.

Properties

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 {(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
  Inverse limit
  In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
  
   is a continuum.

A finite or countable product of continua is a continuum.

See also

• Linear continuum
  Linear continuum
  In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line.Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two members there is another, and which "lacks gaps" in the...
  
  
• Menger sponge
  Menger sponge
  In mathematics, the Menger sponge is a fractal curve. It is a universal curve, in that it has topological dimension one, and any other curve is homeomorphic to some subset of it. It is sometimes called the Menger-Sierpinski sponge or the Sierpinski sponge...
  
  
• Shape theory (mathematics)
  Shape theory (mathematics)
  Shape theory is a branch of the mathematical field of topology. Homotopy theory is not appropriate for spaces with bad local properties, hence the need for replacement of homotopy theory by a more sophisticated approach...
  
  

In the mathematical field of point-set topology, a continuum (pl continua) is a nonempty compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 connected

Connected space

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 nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

 metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

, or less frequently, a compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 connected

Connected space

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 nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

 Hausdorff topological space

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

. Continuum theory is the branch of topology devoted to the study of continua

Definitions

• A continuum that contains more than one point is called nondegenerate.

• A subset A of a continuum X such that A itself a continuum is called a subcontinuum of X. A space homeomorphic to a subcontinuum of the Euclidean plane R² 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
  Indecomposable continuum
  In point-set topology, an indecomposable continuum is a continuum that is not the union of any two of its proper subcontinua. The pseudo-arc is an example of a hereditarily indecomposable continuum. L. E. J...
  
   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.

Examples

• 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.

• Topologist's sine curve
  Topologist's sine curve
  In the branch of mathematics known as topology, the topologist's sine curve is a topological space with several interesting properties that make it an important textbook example....
  
   is a subset of the plane which 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
  Topologist's sine curve
  In the branch of mathematics known as topology, the topologist's sine curve is a topological space with several interesting properties that make it an important textbook example....
  
   by an arc connecting (0,−1) and (1,sin(1)). It is a one-dimensional continuum whose homotopy group
  Homotopy group
  In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
  
  s are all trivial, but it is not a contractible space
  Contractible space
  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....
  
  .

• An n-cell is a space homeomorphic to the closed ball in the Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
   Rⁿ. It is contractible and is the simplest example of an n-dimensional continuum.

• An n-sphere is a space homeomorphic to the standard n-sphere in the (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
  Hilbert cube
  In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology...
  
   is an infinite-dimensional continuum.

• Solenoid
  Solenoid (mathematics)
  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...
  
  s are among the simplest examples of indecomposable homogeneous continua. They are neither arcwise connected nor locally connected.

• Sierpinski carpet
  Sierpinski carpet
  The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions . Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional...
  
  , 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.

• Pseudo-arc
  Pseudo-arc
  In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. Pseudo-arc is an arc-like homogeneous continuum. R.H...
  
   is a homogeneous hereditarily indecomposable planar continuum.

Properties

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 {(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
  Inverse limit
  In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
  
  is a continuum.

A finite or countable product of continua is a continuum.

External links

• 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.