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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. If removal of a point doesn't result in disconnected spaces, this point is called a non-cut point. 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 to determine whether two connected spaces are homeomorphic by counting the number of cut-points in each space. If two spaces have different number of cut-points, they are not homeomorphic. A classic example is using cut-points to show that lines and circles are not homeomorphic. Cut-points are also 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. Definition Formal definitions A cut-point of a connected T1 topological space ''X'', is a point ''p'' in ''X'' such that ''X' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cut-point Line And Circle
In topology, a cut-point is a point of a connected space such that its removal causes the resulting space to be disconnected. If removal of a point doesn't result in disconnected spaces, this point is called a non-cut point. 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 to determine whether two connected spaces are Homeomorphism, homeomorphic by counting the number of cut-points in each space. If two spaces have different number of cut-points, they are not homeomorphic. A classic example is using cut-points to show that lines and circles are not homeomorphic. Cut-points are also useful in the characterization of continuum (topology), topological continua, a class of spaces which combine the properties of compact space, compactness and connected space, connectedness and include many familiar spaces such as the unit interval, the circle, and the torus. Definition Formal definitions A cut-point of a Connected s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
