Examples of Topological Spaces

On The Fundamentals of Topological Spaces we defined what a topological space is gave some basic definitions - including definitions of open sets, closed sets, the interior of a set, and the closure of a set. We will now give some examples of topologies and topological spaces.

The Indiscrete Topology (Trivial Topology)

Definition: Let $X$ be a nonempty set. The Indiscrete Topology or Trivial Topology on $X$ is the topology which contains only $\emptyset$ and $X$, that is, $\tau = \{ \emptyset, X \}$.

The indiscrete topology is fairly trivial (hence it's name). It is easy to verify that $X$ with the indiscrete topology is indeed a topological space.

The Discrete Topology

Definition: Let $X$ be a nonempty set. The Discrete Topology on $X$ is the topology which contains every subset of $X$, that is, $\tau = \mathcal P(X)$.

The Metric Topology

Definition: Let $(X, d)$ be a metric space. The Metric Topology on $X$ is the topology which contains all open balls in $X$.

Note that if $X$ is a normed space then $(X, \| \cdot \|_X)$ is a metric space. The norm topology on $X$ is simply the metric topology on $X$ where the metric $d$ is defined to be $d(x, y) = \| x - y \|_X$.