Section Examples of Topologies
In our preview activity we saw several examples of topologies. Suppose \(X\) is a nonempty set.
The topology consisting of all subsets of \(X\) is called the discrete topology.
The topology \(\{\emptyset, X\}\) is the indiscrete topology.
If \((X,d)\) is a metric space, then the collection consisting of unions of all open balls is a topology called the metric topology. This result tells us that every metric space is topological space under the metric topology. We will see later than not every topological space is a metric space.
The discrete and indiscrete topologies are topologies that can be defined on any set and are often used to use to generate examples. Another topology that can be defined on any set is in the next activity.
Activity 12.2.
Let \(X\) be any set and let \(\tau_{FC}\) consist of the empty set along with all subsets \(O\) of \(X\) such that \(X \setminus O\) is finite.
(a)
Prove that \(\tau_{FC}\) is a topology on \(X\text{.}\) (The topology \(\tau_{FC}\) is called the finite complement topology or the cofinite topology.
(b)
Explain why \(\tau_{FC}\) is the discrete topology when \(X\) is finite.
