General topology/Definition, characterisations

From Wikibooks, open books for an open world
Jump to: navigation, search

The very basics[edit]

Definition (topology):

Let be any set. A topology on is a subset of the power set of such that the following three axioms hold:

  1. and

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 .

Neighbourhoods[edit]

Definition (open neighbourhood):

Let be a topological space, and let . A neighbourhood of is an open set (ie. ) such that .

Definition (neighbourhood):

Let be a topological space, and let A neighbourhood of is a set such that there exists an open set such that and .

Exercises[edit]