$\newcommand{\int}{\operatorname{int}}$If $\langle X,\tau\rangle$ is a topological space, $\langle\tau,\subseteq\rangle$ is a complete distributive lattice: for any $\mathscr{U}\subseteq\tau$, $\bigvee\mathscr{U}=\bigcup\mathscr{U}$, and $\bigwedge\mathscr{U}=\int\bigcap\mathscr{U}$, and for any $U,V,W\in\tau$ we have $U\land(V\lor W)=U\cap(V\cup W)=(U\cap V)\cup(U\cap W)=(U\land V)\lor(U\land W)$ and $U\lor(V\land W)=U\cup(V\cap W)=(U\cup V)\cap(U\cup W)=(U\lor V)\land(U\lor W)\;.$ What distinguishes the two is infinite distributivity: for any $V\in\tau$
$V\land\bigvee\mathscr{U}=V\cap\bigcup\mathscr{U}=\bigcup_{U\in\mathscr{U}}(V\cap U)=\bigvee_{U\in\mathscr{U}}(V\land U)\;,$
but it’s not true in general that
$V\lor\bigwedge\mathscr{U}=V\cup\int\bigcap\mathscr{U}\overset{?}=\int\bigcap_{U\in\mathscr{U}}(V\cup U)=\bigwedge_{U\in\mathscr{U}}(V\lor U)\;.$
What you really want to talk about, I suspect, are frames; see also pointless topology (a name that I occasionally find regrettably apt!).