 Continuum (topology)

In the mathematical field of pointset topology, a continuum (pl continua) is a nonempty compact connected metric space, or less frequently, a compact connected Hausdorff topological space. Continuum theory is the branch of topology devoted to the study of continua
Contents
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 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 onedimensional 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 onedimensional, arcwise connected, and locally connected.
 Topologist's sine curve 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 yaxis. It is a onedimensional continuum that is not arcwise connected, and it is locally disconnected at the points along the yaxis.
 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 onedimensional continuum whose homotopy groups are all trivial, but it is not a contractible space.
 An ncell is a space homeomorphic to the closed ball in the Euclidean space R^{n}. It is contractible and is the simplest example of an ndimensional continuum.
 An nsphere is a space homeomorphic to the standard nsphere in the (n + 1)dimensional Euclidean space. It is an ndimensional homogeneous continuum that is not contractible, and therefore different from an ncell.
 The Hilbert cube is an infinitedimensional continuum.
 Solenoids are among the simplest examples of indecomposable homogeneous continua. They are neither arcwise connected nor locally connected.
 Sierpinski carpet, also known as the Sierpinski universal curve, is a onedimensional planar Peano continuum that contains a homeomorphic image of any onedimensional planar continuum.
 Pseudoarc 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 is a continuum.
A finite or countable product of continua is a continuum.
See also
References
 Sam B. Nadler, Jr, Continuum theory. An introduction. Pure and Applied Mathematics, Marcel Dekker. ISBN 0824786599
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. (NorthHolland), 1990.
Categories: Continuum theory
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