any collection $\col{U}$ of open sets exists via $\bigvee \col{U} = \bigcup \col{U}$. We note that a poset in which all infima and suprema exist for finite collections is usually called a *lattice*, so every topological space defines a lattice of open sets.

Although the Definitions above suggest that we can merely focus on posets, avoiding mention of topological spaces, this is only partially true. If we take a given poset as ${\bf Open}(X,\col{T})$, this alone does not completely define a topological space.

\[ex:same\_poset\] Consider the set $X=\{a,b,c\}$ with two topologies, $$\col{T}_1 = \{\{a,b,c\}, \{a,b\},\{c\},\emptyset\}$$ and $$\col{T}_2=\{\{a,b,c\},\{a\},\{c\},\emptyset\}.$$ Both of these topologies have the same poset of open sets, namely ${\bf Open}(X,\col{T}_1)$ at left below and ${\bf Open}(X,\col{T}_2)$ at right $$\xymatrix{
&\emptyset\ar[dl]\ar[dr]& & &\emptyset\ar[dl]\ar[dr]&\\
\{a,b\}\ar[dr]&&\{c\}\ar[dl] & \{a\}\ar[dr] &&\{c\}\ar[dl]\\
&\{a,b,c\}& & &\{a,b,c\}\\
}$$ Yet $(X,\col{T}_1)$ and $(X,\col{T}_2)$ are quite different as topological spaces. Observe that both $\{a,b\} \vee \{c\} = \{a,b,c\}$ in ${\bf Open}(X,\col{T}_1)$ and $\{a\} \vee \{c\} = \{a,b,c\}$ in ${\bf Open}(X,\col{T}_2)$. However, only in $\col{T}_1$ is the union of these two elements $\{a,b\} \cup \{c\} = \{a,b,c\}$.

One particularly important topology that can be built from a partial order is the *Alexandroff* topology.

[@Alexandroff_1937] \[df:alexandroff\] In a poset $(P,\le)$, the 