# Definition:Topology

## Definition

### Definition 1

Let $S$ be a set.

A **topology on $S$** is a subset $\tau \subseteq \powerset S$ of the power set of $S$ that satisfies the open set axioms:

\((\text O 1)\) | $:$ | The union of an arbitrary subset of $\tau$ is an element of $\tau$. | |||||||

\((\text O 2)\) | $:$ | The intersection of any two elements of $\tau$ is an element of $\tau$. | |||||||

\((\text O 3)\) | $:$ | $S$ is an element of $\tau$. |

If $\tau$ is a **topology** on $S$, then $\struct {S, \tau}$ is called a topological space.

The elements of $\tau$ are called the open sets of $\struct {S, \tau}$.

### Definition 2

Let $S$ be a set.

A **topology on $S$** is a subset $\tau \subseteq \powerset S$ of the power set of $S$ that satisfies the following axioms:

\((\text O 1')\) | $:$ | The union of an arbitrary subset of $\tau$ is an element of $\tau$. | |||||||

\((\text O 2')\) | $:$ | The intersection of any finite subset of $\tau$ is an element of $\tau$. |

## Also defined as

Some sources include the additional assumption that $\O \in \tau$.

This is redundant as established in Empty Set is Element of Topology, since $(\text O 2)$ and $(\text O 1')$ are read to include the union over the empty set, which is equal to $\O$.

## Empty Topological Space

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.

## Also see

- $\sigma$-algebra, which looks similar on the surface to a topology, but closed (in the algebraic sense) under countable unions. A topology has no such limitation on countability.

- Results about
**topologies**can be found**here**.

## Linguistic Note

The word **topology** is derived from the Greek words **τόπος**, meaning **place**, and **λόγος**, meaning **study**.

This originates from its origins as the study of *situation*.

The word **topology** was coined by Johann Benedict Listing in a letter of $1836$.

The word was first published in the title of Listing's $1847$ book *Vorstudien zur Topologie*, but never caught on until the $1920$s.