# General topology/Constructions

## Lattice of topologies on a fixed set

Proposition (topologies on a fixed set are a complete lattice):

Let ${\displaystyle X}$ be a set, and let ${\displaystyle (\tau _{i})_{i\in I}}$ be a family of topologies on ${\displaystyle X}$. Then there exists a

## Final and initial topologies

Proposition (preimage of a topology is a topology):

Let ${\displaystyle f:X\to Y}$ be a function from a set ${\displaystyle X}$ to another set ${\displaystyle Y}$, and let ${\displaystyle \tau }$ be a topology on ${\displaystyle Y}$. Then ${\displaystyle f^{-1}(\tau )}$ is a topology on ${\displaystyle X}$.

Proof: This follows instantly from the formulae

1. ${\displaystyle f^{-1}(Y)=X}$, ${\displaystyle f^{-1}(\emptyset )=\emptyset }$
2. ${\displaystyle f^{-1}\left(\bigcup _{i\in I}B_{i}\right)=\bigcup _{i\in I}f^{-1}(B_{i})}$ and ${\displaystyle f^{-1}\left(\bigcap _{i\in I}B_{i}\right)=\bigcap _{i\in I}f^{-1}(B_{i})}$ for any family of subsets ${\displaystyle (B_{i})_{i\in I}}$ of ${\displaystyle Y}$
${\displaystyle \Box }$

Definition (initial topology):

Let ${\displaystyle X}$ be a set and let ${\displaystyle (X_{i},\tau _{i})_{i\in I}}$ be a family of topological spaces. Let further ${\displaystyle f_{i}:X\to X_{i}}$ (${\displaystyle i\in I}$) be functions. The initial topology on ${\displaystyle X}$