Half-disk topology

In mathematics, and particularly general topology, the half-disk topology is an example of a topology given to the set [math]\displaystyle{ X }[/math], given by all points [math]\displaystyle{ (x,y) }[/math] in the plane such that [math]\displaystyle{ y\ge 0 }[/math]. The set [math]\displaystyle{ X }[/math] can be termed the closed upper half plane. To give the set [math]\displaystyle{ X }[/math] a topology means to say which subsets of [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ X }[/math] and the empty set [math]\displaystyle{ \emptyset }[/math] are open sets.

Construction

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

See also

• List of topologies