Half-disk topology

In mathematics, and particularly general topology, the half-disk topology is an example of a topology given to the set X, given by all points (x,y) in the plane such that y ≥ 0.^[1] The set X can be termed the closed upper half plane.

To give the set X a topology means to say which subsets of X 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 X and the empty set ∅ are open sets.

Construction

We consider X to consist of the open upper half plane P, given by all points (x,y) in the plane such that y > 0; and the x-axis L, given by all points (x,y) in the plane such that y = 0. Clearly X is given by the union P ∪ L. The open upper half plane P has a topology given by the Euclidean metric topology.^[1] We extend the topology on P to a topology on X = P ∪ L by adding some additional open sets. These extra sets are of the form {(x,0)} ∪ {P ∩ U}, where (x,0) is a point on the line L and U is an open, with respect to the Euclidean metric topology, neighbourhood of (x,y) in the plane.^[1]

References

1. 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