A topological space related to SAT: is it compact?
What you are doing is deriving a topological representation of a Boolean algebra. The study of representations of Boolean algebras goes back at least to Lindenbaum and Tarski who proved (in 1925, I think) that the complete, atomic Boolean algebras are isomorphic to powerset lattices. 

There are however, Boolean algebras that are not complete and atomic. For example, the sequence $x_1, x_1 \land x_2, \ldots$, is a descending chain that has no limit in the Boolean algebra defined over formulas. The question of whether arbitrary Boolean algebras, such as the one you mention, also had set-based representations was solved by Marshall Stone, who put forth the maxim "always topologize" (Marshall H. Stone. The representation of Boolean algebras, 1938).


  Stone's Representation Theorem for Boolean Algebras Every Boolean algebra is isomorphic to the lattice of clopen subsets of a topological space.


The main idea is to consider what in your case are the satisfying assignments to a formula. In the general case, you consider  homomorphisms from a Boolean algebra into the two element Boolean algebra (the truth values). The inverse of $\mathit{true}$ gives you the sets of satisfying assignments, or what are called ultrafilters of the Boolean algebra. From these, one can obtain a topology called the spectrum or Stone space of a Boolean algebra. Stone provide the answer to your question too.


  The Stone space of a  Boolean algebra is a compact, totally disconnected Hausdorff space.  


There have been several results that extend and generalise Stone's representation in various directions. A natural question is to ask if other families of lattices have such representations. Stone's results also apply to distributive lattices. Topological representations for arbitrary lattices were given by Alasdair Urquhart in 1978. Distributive lattices enjoy greater diversity in structure, compared to Boolean algebras and are of great interest. A different  representation for the distributive case was given by Hilary Priestley in 1970, using the idea of an ordered topological space. Instead of set-based representations, we can find poset-based representations and topologies.

The constructions in these papers have one remarkable property. Stone's construction maps not just Boolean algebras to topological spaces: structural relationships relating Boolean algebras translate into structural properties between the resulting topologies. It is a duality between categories. The entire gamut of such results is called Stone Duality. Informally, dualities give us precise translations between mathematical universes: the combinatorial world of sets, the algebraic world of lattices, the spatial world of topology and the deductive world of logic. Here are a few starting points that may help.


Chapter 11 of Introduction to Lattices and Order, by Davey and Priestley covers Stone's theorem.
Matthew Gwynne's slides  cover the theorem and give a proof of compactness. Matthew (in the comments) also suggests Introduction to Boolean Algebras by Paul Halmos.
In moving from propositional logic to modal logic, the Boolean algebra is extended with a join-preserving operator and topology with an interior.  Jónsson and Tarski's 1952 paper, Boolean Algebras with Operators is extremely readable and consistent with modern notation. 
Chapter 5 of Modal Logic by Blackburn, de Rijke and Venema covers Stone's theorem and its extension to Boolean algebras with operators.
Stone Spaces by Peter Johnstone reviews such results for various other kinds of algebras.