Topology and topological space. Definition
A topology on a set
- T1
- Whenever
is a family (finite or not) of subsets of such that for all , then . - T2
- Whenever
, then . - T3
and .
A topological space
T3 can be derived from T1 and T2, and is not strictly necessary as an axiom; however, most treatments of the subject introduce the definition as this triplet of statements.[really?]
In relaxed words
T1 can be phrased as 'an arbitrary union of open subsets is open'. By induction T2 can be phrased as 'a finite intersection of open subsets is open'.
Combining these two then, a topology on
Example
Discrete and indiscrete topologies
The discrete and indiscrete topologies are two important topologies
(which can be defined for any set). With
- the collection of all subsets of
is a topology on ; it is called the discrete topology. - the collection of subsets of
consisting of and is a topology on ; it is called the indiscrete topology.
The name 'discrete' is intended to suggest indivisibility of parts—it is not possible to arrive at a new topology by dividing elements of the discrete topology into smaller subsets (those smaller subsets must already be in the discrete topology).
Finite complement topology
Let
Countable complement topology
A similar construction to the finite complement topology, the
countable complement topolgy is the collection of subsets
of a set