Continuum (topology) · Lablypedia

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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(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.

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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.

A component of a particular compact Hausdorff space can be formulated in terms of continua; it is defined as a continuum that contains two arbitrary points of a compact and Hausdorff 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 (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 {(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.