In mathematics, and particularly general topology, the half-disk topology is an example of a topology given to the set , given by all points in the plane such that .[1] The set can be termed the closed upper half plane.

To give the set a topology means to say which subsets of are "open", and to do so in a way that the following axioms are met:[2]

  1. The union of open sets is an open set.
  2. The finite intersection of open sets is an open set.
  3. The set and the empty set are open sets.

Construction

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We consider   to consist of the open upper half plane  , given by all points   in the plane such that  ; and the x-axis  , given by all points   in the plane such that  . Clearly   is given by the union  . The open upper half plane   has a topology given by the Euclidean metric topology.[1] We extend the topology on   to a topology on   by adding some additional open sets. These extra sets are of the form  , where   is a point on the line   and   is a neighbourhood of   in the plane, open with respect to the Euclidean metric (defining the disk radius).[1]

See also

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References

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  1. ^ a b c Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, pp. 96–97, ISBN 0-486-68735-X
  2. ^ Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, p. 3, ISBN 0-486-68735-X