A **topological space** is a pair $(X,\mathcal{O}(X))$ of a set $X$ and a set $\mathcal{O}(X)$ of subsets of $X$. We call the members of $\mathcal{O}(X)$ **open sets** and it satisfies: 1. $X \in \mathcal{O}(X)$, 2. $\emptyset \in \mathcal{O}(X)$, 3. If $U,V \in \mathcal{O}(X)$ then $U \cap V \in \mathcal{O}(X)$, 4. If $U_i \in \mathcal{O}(X)$ then $\bigcup U_i \in \mathcal{O}(X)$. We will often simply write $X$ and use $\mathcal{O}(X)$ when we must refer to its opens. A topological space can be viewed as a model of the logic of [[Verifiability]]. The correct notion of a morphism between two topological spaces is a [[Continuous Function]]. The set $\mathcal{O}(X)$ is a [[Lattice]] and in particular a [[Frame]].