Half-disk topology

In mathematics, and particularly general topology, the half-disk topology is an example of a topology given to the set formula_1, given by all points formula_2 in the plane such that formula_3. The set formula_1 can be termed the closed upper half plane.
To give the set formula_1 a topology means to say which subsets of formula_1 are "open", and to do so in a way that the following axioms are met:
Construction.
We consider formula_1 to consist of the open upper half plane formula_10, given by all points formula_2 in the plane such that formula_12; and the "x"-axis formula_13, given by all points formula_2 in the plane such that formula_15. Clearly formula_1 is given by the union formula_17. The open upper half plane formula_10 has a topology given by the Euclidean metric topology. We extend the topology on formula_10 to a topology on formula_20 by adding some additional open sets. These extra sets are of the form formula_21, where formula_22 is a point on the line formula_13 and formula_24 is an open, with respect to the Euclidean metric topology, neighbourhood of formula_22 in the plane.