# General topology/Definition, characterisations

## The very basics

Definition (topology):

Let ${\displaystyle X}$ be any set. A topology on ${\displaystyle X}$ is a subset ${\displaystyle \tau \subset {\mathcal {P}}(X)}$ of the power set of ${\displaystyle X}$ such that the following three axioms hold:

1. ${\displaystyle X\in \tau }$ and ${\displaystyle \emptyset \in \tau }$
2. ${\displaystyle U_{1},\ldots ,U_{n}\in \tau \Rightarrow U_{1}\cap \cdots \cap U_{n}\in \tau }$
3. ${\displaystyle (\forall i\in I:U_{i}\in \tau )\Rightarrow \bigcup _{i\in I}U_{i}\in \tau }$

The set within a topology are called the open sets of ${\displaystyle (X,\tau )}$.

Proposition (definition of topologies from closed sets):

Let ${\displaystyle X}$ be a set, and let

Example (Euclidean topology):

Let ${\displaystyle n\in \mathbb {N} }$ and consider the set ${\displaystyle X=\mathbb {R} ^{n}}$. Then we define

Proposition (topologies are ordered by inclusion):

If ${\displaystyle X}$ is any set and ${\displaystyle \tau _{1},\tau _{2}}$ are topologies on ${\displaystyle X}$, then ${\displaystyle \tau _{1}\leq \tau _{2}:\Leftrightarrow \tau _{1}\subseteq \tau _{2}}$ defines an order on the topologies of ${\displaystyle X}$.

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

${\displaystyle \Box }$

Definition (comparison of topologies):

Let ${\displaystyle X}$ be any set and ${\displaystyle \tau _{1},\tau _{2}}$ topologies on ${\displaystyle X}$.

## Neighbourhoods

Definition (open neighbourhood):

Let ${\displaystyle (X,\tau )}$ be a topological space, and let ${\displaystyle x\in X}$. A neighbourhood of ${\displaystyle x}$ is an open set ${\displaystyle U}$ (ie. ${\displaystyle U\in \tau }$) such that ${\displaystyle x\in U}$.

Definition (neighbourhood):

Let ${\displaystyle (X,\tau )}$ be a topological space, and let ${\displaystyle x\in X}$ A neighbourhood of ${\displaystyle x}$ is a set ${\displaystyle N\subset X}$ such that there exists an open set ${\displaystyle U\in \tau }$ such that ${\displaystyle U\subseteq N}$ and ${\displaystyle x\in U}$.