Topology
Overview
Topology is the branch of mathematices that deals with formalizing the notion of "closeness", and then the related notion
of a continuous transformation. (That is, a continuous transformation is one that sends points close to each other to points
that are close to each other)
Early notions of closeness used the notion of a
metric, that is a function that specified the distance between two points. (see below and
metric topology)
Closeness in its most abstract form is defined by defining the notion of an open set.
Axioms of a Topology
A topology on a
set
X is a collection of subsets Y such that
- if {% A \in Y \; and \; B \in Y \rightarrow A \cap B \in Y %}
- the union of any collection of members of Y is in Y
- {% \emptyset \in Y \; and \; Y \in Y %}
The members of the set Y are called
open sets.
A
closed set is a set whose
complement
is open.
Inducing a Topology
Even though a topology is defined as a collection of open sets, it is often common to define the idea of an open set from some
other notion defined on the space first.
Metric Topology - the most common way to define an open set from some prior
notions.
Topological Notions
Once the idea of closeness has been defined, other notions such as continuity follow from it.
Basis of a Topology
A
basis
of a topology is a set of sets from which the whole topology can be constructed.
