Half-disk topology

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

To give the set ${\displaystyle X}$ a topology means to say which subsets of ${\displaystyle X}$ 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 ${\displaystyle X}$ and the empty set ${\displaystyle \emptyset }$ are open sets.