Toplogy, a Categorical view

Consider a topology space $(X,\tau )$$(X,\tau)$

We can view it as a Category, ${\mathrm{Ob}}_{(X,\tau )}=\tau ,{\mathrm{Hom}}_{(X,\tau )}=\subseteq$$\mathrm{Ob}_{(X,\tau)}=\tau,\mathrm{Hom}_{(X,\tau)}=\subseteq$

The first axiom means the category has the initial and final object $\mathrm{\varnothing },X$$\empty,X$

The second axiom means the category is closed under infinite coproduct

$\begin{array}{c}\coprod _{i=1}^{\mathrm{\infty }}{O}_i=\bigcup _{i=1}^{\mathrm{\infty }}{O}_i\in \tau \end{array}$$\coprod_{i=1}^\infty O_i=\bigcup_{i=1}^\infty O_i\in\tau\\$

The third axiom means the category is closed under finite product

$\begin{array}{c}\prod _{i=1}^n{O}_i=\bigcap _{i=1}^n{O}_i\end{array}$$\prod_{i=1}^n O_i=\bigcap_{i=1}^n O_i\\$

And the complement gives us a $\mathrm{functor}$$\mathrm{functor}$ $(O_i\subseteq O_j{)}^c\iff O_j^c\subseteq O_i^c$$(O_i\subseteq O_j)^c\Leftrightarrow O_j^c\subseteq O_i^c$

${\mathrm{\varnothing }}^c=X,X^c=\mathrm{\varnothing }$$\empty^c=X,X^c=\empty$

$\begin{array}{c}{\left(\coprod _{i=1}^{\mathrm{\infty }}{O}_i\right)}^c=\bigcap _{i=1}^{\mathrm{\infty }}{O}_i^c=\prod _{i=1}^{\mathrm{\infty }}{O}_i^c\end{array}$$\left(\coprod_{i=1}^\infty O_i\right)^c=\bigcap_{i=1}^\infty O_i^c=\prod_{i=1}^\infty O_i^c\\$

$\begin{array}{c}{\left(\prod _{i=1}^n{O}_i\right)}^c=\bigcup _{i=1}^n{O}_i^c=\coprod _{i=1}^n{O}_i\end{array}$$\left(\prod_{i=1}^n O_i\right)^c=\bigcup_{i=1}^n O_i^c=\coprod_{i=1}^n O_i\\$

This functor provides a way to define topology using closed sets.