Английская Википедия:Half-disk topology

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

In mathematics, and particularly general topology, the half-disk topology is an example of a topology given to the set <Math>X</Math>, given by all points <Math>(x,y)</Math> in the plane such that <Math>y\ge 0</Math>. The set <Math>X</Math> can be termed the closed upper half plane.

To give the set <Math>X</Math> a topology means to say which subsets of <Math>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>X</Math> and the empty set <Math>\emptyset</Math> are open sets.

Construction

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

See also

• List of topologies