Trivial topology

Definition

The trivial topology (sometimes known as the indiscrete topology)is an example of a topological space that exists for any set $X$, it is defined as follows:

• Given a set $X$ we define the open sets as $\mathcal{J}:=\{\emptyset,X\}$

Then $(X,\mathcal{J})$ is a topology.

Contrast to the Discrete topology

There is at least 1 other topology that can be defined on an arbitrary set, the Discrete topology, which is a topology induced by a metric, the Discrete metric specifically.
Warning, the following is Alec's speculation

Unlike the discrete topology the indiscrete, or trivial topology is not induced by a metric. For if such a metric existed it would have to have the open ball of radius $0$ as the entire of $X$, then no strict :subset of $X$ (except the emptyset) is a neighborhood to all of its points, thus not open.

However this is not a metric as the metric must assign two points a distance of $0$ if and only if they are the same point.

I'm not entirely happy about this proof, however there is logic here! I will do this formally later.

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TODO: Formally prove this

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End of warning