# General topology/Definition, characterisations

## The very basics[edit]

**Definition (topology)**:

Let be any set. A **topology** on is a subset of the power set of such that the following three axioms hold:

- and

The set within a topology are called the **open sets** of .

**Proposition (definition of topologies from closed sets)**:

Let be a set, and let

**Example (Euclidean topology)**:

Let and consider the set . Then we define

**Proposition (topologies are ordered by inclusion)**:

If is any set and are topologies on , then defines an order on the topologies of .

**Proof:** This is a special case of the general fact that Order theory/

**Definition (comparison of topologies)**:

Let be any set and topologies on .

## Neighbourhoods[edit]

**Definition (open neighbourhood)**:

Let be a topological space, and let . A **neighbourhood** of is an open set (ie. ) such that .

**Definition (neighbourhood)**:

Let be a topological space, and let A **neighbourhood** of is a set such that there exists an open set such that and .