Definition 3.1.1 (Topological Space).label Let $X$ be a set, possibly empty. A topology on $X$ is a structure given by a collection $\tau\suf\mathcal{P}(X)$ satisfying:

1. (O1)

   $\emp,X\in \tau$

2. (O2)

   $\fall \curl{U_i}_{i\in I}\suf \tau,\cups{}{i\in I}U_{i}\in \tau$

3. (O3)

   $\fall U,V\in \tau, U\cap V\in \tau$

The ordered pair $(X,\tau)$ (or just $X$ if $\tau$ is understood) is a topological space and elements of $\tau$ are open sets.

The elements of a topological space are points and $X$ is the underlying set of the topological space $(X,\tau)$. If $\tau\suf \tau'$ are topologies on $X$ then $\tau$ is courser than $\tau'$ and $\tau'$ is finer than $\tau$.