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 $$y\ge 0$$. 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:


 * 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 $$\emptyset$$ 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 $$P\cup L$$. The open upper half plane $$P$$ has a topology given by the Euclidean metric topology. We extend the topology on $$P$$ to a topology on $$X=P\cup L$$ by adding some additional open sets. These extra sets are of the form $${(x,0)}\cup (P\cap U)$$, where $$(x,0)$$ is a point on the line $$L$$ and $$U$$ is a neighbourhood of $$(x,0)$$ in the plane, open with respect to the Euclidean metric (defining the disk radius).