Half-disk topology
Encyclopedia
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 . The set X can be termed the closed upper half plane.
To give the set X a topology means to say which subset
s of X are "open", and to do so in a way that the following axiom
s are met:
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 by adding some additional open sets. These extra sets are of the form where (x,0) is a point on the line L and U is an open, with respect to the Euclidean metric topology, neighbourhood of (x,y) in the plane.
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
, the half-disk topology is an example of a topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
given to the set X, given by all points (x,y) in the plane such that . The set X can be termed the closed upper half plane.
To give the set X a topology means to say which subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of X are "open", and to do so in a way that the following axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s are met:
- The union of open sets is an open set.
- The finite intersection of open sets is an open set.
- The set X and the empty setEmpty setIn mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
∅ 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 ; and the x-axis L, given by all points (x,y) in the plane such that . Clearly X is given by the unionUnion (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
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 by adding some additional open sets. These extra sets are of the form where (x,0) is a point on the line L and U is an open, with respect to the Euclidean metric topology, neighbourhood of (x,y) in the plane.