The very basics
Let be any set. A topology on is a subset of the power set of such that the following three axioms hold:
The set within a topology are called the open sets of .
Proposition (definition of topologies from closed sets):
Let be a set, and let
Example (Euclidean topology):
Let and consider the set . Then we define
Proposition (topologies are ordered by inclusion):
If is any set and are topologies on , then defines an order on the topologies of .
Proof: This is a special case of the general fact that Order theory/
Definition (comparison of topologies):
Let be any set and topologies on .
Definition (open neighbourhood):
Let be a topological space, and let . A neighbourhood of is an open set (ie. ) such that .
Let be a topological space, and let A neighbourhood of is a set such that there exists an open set such that and .