Continuum (topology)

Definitions

• 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 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 = 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.

Examples

• An arc is a space homeomorphic to the closed interval . If h: → X is a homeomorphism and h = p and h = 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 = sin, 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 and. It is a one-dimensional continuum whose homotopy groups are all trivial, but it is not a contractible space.
• An n-cell is a space homeomorphic to the closed ball in the Euclidean space 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 -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.

Properties