examples of topological space

       Examples of topological spaces

For any set , there are two topologies we can always define on :

• The Discrete topology - the topology consisting of all subsets of a set .
• The Indiscrete topology (also known as the trivial topology) - the topology consisting of just  and the empty set, .

Metric Topology

Given a metric space , its metric topology is the topology induced by using the set of all open balls as the base. One can also define the topology induced by the metric, as the set of all open subsets defined by the metric. We denote the topology  induced from the metric d with 

This forms a topological space from a metric space.

If for a topological space , we can find a metric , such that , then the topological space is called metrizable.

The usual topology on the real numbers

We can define a topology  on  by defining  to be in  if for every point , there is an  such that . We call this topology the standard topology, or usual topology on .

The cofinite topology on any set

Let  be a non-empty set. Define  to be the collection of all subsets  of satisfying the following:

1. Either 
2. Or  is finite.

Then  is a topology on  called the cofinite topology (or "finite complement topology") on . Further, this topology turns out to be discrete if and only if is finite.

The cocountable topology on any set

Let  be a non-empty set. Define  to be the collection of all subsets  of satisfying the following:

1. Either 
2. Or  is countable.

Then  is a topology on  called the cocountable topology (or "countable complement topology") on . Further, this topology turns out to be discrete if and only if  is countable.