Definition:Topological Space

From ProofWiki
Jump to: navigation, search

Definition

Let $S$ be a set.

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

That is, let $\tau \subseteq \mathcal P \left({S}\right)$ satisfy the open set axioms:

\((O1)\)   $:$   The union of an arbitrary subset of $\tau$ is an element of $\tau$.             
\((O2)\)   $:$   The intersection of any two elements of $\tau$ is an element of $\tau$.             
\((O3)\)   $:$   $S$ is an element of $\tau$.             


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.


Can a topological space be empty?

Notwithstanding the result Empty Set Satisfies Topology Axioms, it is frequently stipulated in the literature that the class of topological spaces does not include the empty set.

This convention is sufficiently commonplace as to be often omitted in published texts, and taken for granted. When it is mentioned, it is usually given as an afterthought.


However, there exists a philosophical position that disallowing the empty topological space is unhelpful, and even harmful.

$\mathsf{Pr} \infty \mathsf{fWiki}$ adopts this philosophical position, allowing that the underlying set of a given topological space may indeed be empty.


Note, however, that many of the possible properties of a topological space are held vacuously by the empty space. This position having been taken, it is necessary in many cases to add a condition to a given general statement made about spaces specifically to exclude the empty space from the scope of that statement.


Sources