Metric and Topological Spaces

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## Definition and examples of topologies

We now build on the idea of "open sets" introduced earlier.

Definition
Let X be a set. A set of subsets of X is called a topology (and the elements of are called open sets) if the following properties are satisfied.

1. (the empty set), X  ,
2. if {Ai | i I}  then Ai  ,
3. if A, B  then A B  .

Remarks
1. Conditions 2. and 3. can be summarised as:
The topology is closed under arbitrary unions and finite intersections.
2. (X, ) is called a topological space.

Examples
1. The prototype
Let X be any metric space and take to be the set of open sets as defined earlier. The properties verified earlier show that is a topology.

2. Some "extremal" examples
Take any set X and let = { , X}. Then is a topology called the trivial topology or indiscrete topology.
Let X be any set and let be the set of all subsets of X. The is a topology called the discrete topology. It is the topology associated with the discrete metric.

Remark
A topology with many open sets is called strong; one with few open sets is weak.
The discrete topology is the strongest topology on a set, while the trivial topology is the weakest.

3. Finite examples
Finite sets can have many topologies on them.
For example, Let X = {a, b} and let ={ , X, {a} }.
Then is a topology called the Sierpinski topology after the Polish mathematician Waclaw Sierpinski (1882 to 1969).

Let X = {1, 2, 3} and = { , {1}, {1, 2}, X}. Then is a topology.

Remark
It is easy to check that the only metric possible on a finite set is the discrete metric. Hence these last two topologies cannot arise from a metric.

4. The Zariski topology
Let X be any infinite set. Define a topology on X by A  if X - A is finite or A = .
This is called the cofinite or Zariski topology after the Belarussian mathematician Oscar Zariski (1899 to 1986)
Examples like this are important in a subject called Algebraic Geometry.

5. A 'different' topology on R
Let X = R and let = { , R} { (x, ) | x R}
Then is a topology in which, for example, the interval (0, 1) is not an open set.
All the sets which are open in this topology are open in the usual topology. That is, this topology is weaker than the usual topology.

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JOC February 2004