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.

Construction

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