Definition:Topological Space

Definition
Let $S$ be any set such that $S \ne \varnothing$.

Let $\tau$ be a topology on $S$.

That is, let $\tau$ satisfy the open set axioms:

Then the ordered pair $\left({S, \tau}\right)$ is called a topological space.

The elements of $\tau$ are called open sets of $\left({S, \tau}\right)$.

In a topological space $\left({S, \tau}\right)$, we consider $S$ to be the universal set.

Also known as
The topological space $\left({S, \tau}\right)$ can be referred to as just a space if the context is clear.

$\left({S, \tau}\right)$ can be referred to as the space $S$ if it is clear what topology is actually carried on it.

Also denoted as
Some authors use the suboptimal $\left\{{S, \tau}\right\}$, which leaves it conceptually unclear as to which is the set and which the topology. This adds unnecessary complexity to the underlying axiomatic justification for the existence of the very object that is being defined.

Note
Notwithstanding the result Empty Set Satisfies Topology Axioms, the stipulation that $S \ne \varnothing$ is standard in the literature.

However, this condition is often omitted in published texts, and taken for granted. When it is mentioned, it is usually given as an afterthought.

Despite the fact that some commentators laughably state, usually without evidence, that disallowing the empty topological space is "harmful", this website will follow tradition and always assume that the condition $S \ne \varnothing$ holds.