A continuum that contains more than one point is callednondegenerate.
A subsetA of a continuumX such thatA itself is a continuum is called asubcontinuum ofX. A space homeomorphic to a subcontinuum of theEuclidean planeR2 is called aplanar continuum.
A continuumX ishomogeneous if for every two pointsx andy inX, there exists a homeomorphismh:X →X such thath(x) =y.
APeano continuum is a continuum that islocally connected at each point.
Anindecomposable continuum is a continuum that cannot be represented as the union of two proper subcontinua. A continuumX ishereditarily indecomposable if every subcontinuum ofX is indecomposable.
Thedimension of a continuum usually means itstopological dimension. A one-dimensional continuum is often called acurve.
Anarc is a spacehomeomorphic to theclosed interval [0,1]. Ifh: [0,1] →X is a homeomorphism andh(0) =p andh(1) =q thenp andq are called theendpoints ofX; one also says thatX is an arc fromp toq. An arc is the simplest and most familiar type of a continuum. It is one-dimensional,arcwise connected, and locally connected.
Thetopologist's sine curve is a subset of the plane that is the union of the graph of the functionf(x) = sin(1/x), 0 <x ≤ 1 with the segment −1 ≤y ≤ 1 of they-axis. It is a one-dimensional continuum that is not arcwise connected, and it is locally disconnected at the points along they-axis.
Ann-cell is a space homeomorphic to the closedball in theEuclidean spaceRn. It is contractible and is the simplest example of ann-dimensional continuum.
Ann-sphere is a space homeomorphic to the standardn-sphere in the (n + 1)-dimensional Euclidean space. It is ann-dimensional homogeneous continuum that is not contractible, and therefore different from ann-cell.
TheHilbert cube is an infinite-dimensional continuum.
Solenoids are among the simplest examples of indecomposable homogeneous continua. They are neither arcwise connected nor locally connected.
TheSierpinski carpet, also known as theSierpinski universal curve, is a one-dimensional planar Peano continuum that contains a homeomorphic image of any one-dimensional planar continuum.
Thepseudo-arc is a homogeneous hereditarily indecomposable planar continuum.
There are two fundamental techniques for constructing continua, by means ofnested intersections andinverse limits.
If {Xn} is a nested family of continua, i.e.Xn ⊇Xn+1, then their intersection is a continuum.
If {(Xn,fn)} is an inverse sequence of continuaXn, called thecoordinate spaces, together withcontinuous mapsfn:Xn+1 →Xn, called thebonding maps, then itsinverse limit is a continuum.
A finite or countable product of continua is a continuum.
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.
