CUT-POINT

In topology, a 'cut-point' is a point of a connected space such that its removal causes the resulting space to be disconnected. For example every point of a line is a cut-point, while no point of a circle is a cut-point. Cut-points are useful in the characterization of topological continua, a class of spaces which combine the properties of compactness and connectedness and include many familiar spaces such as the unit interval, the circle, and the torus.

Contents

Definition

Properties

References

Definition

A 'cut-point' of a connected T₁ topological space ''X'', is a point ''p'' in ''X'' such that ''X'' - {''p''} is not connected. A point which is not a cut-point is called a 'noncut-point'.

Properties

★ Cut-points are not necessarily preserved under continuous functions, (example: ''f'': [0, 2π] → 'R'², given by ''f''(''x'') = (cos ''x'', sin ''x'')), but are preserved under homeomorphisms.

★ Every compact connected Hausdorff space, with more than one point, has at least two noncut-points.

★ Every compact connected metric space, with exactly two noncut-points is homeomorphic to the unit interval.

References

★ General Topology, Willard, Stephen, , , Dover Publications, 2004, ISBN 0486434796