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Let $V$ be the set of truth valuations in propositional logic, and let $\Delta \subseteq PROP$. We define $$[\Delta] = \{v \in V : v(\Delta)=1\}$$. (If $\Delta =\varphi$, then we can write $[\Delta] =[\varphi]$).

I already know that the set $$B =\{ [\varphi] : \varphi \in PROP \} $$ form a basis for a topology $\tau$ on $V$, and that the sets of the form $[\varphi]$ are clopen in $\tau$.

The problem is: prove that the Compactness theorem for propositional logic is equivalent to the topological compactness of $(V,\tau)$.

There're also two hints: (1) prove that the closed sets are exactly those of the form $[\Delta]$; (2) use the characterisation of topological compactness given in terms of closed sets.

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    What is your formulation for the compactness theorem for propositional logic? – 2017-01-09
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    Theorem: let $\Gamma \subseteq PROP$. Then $\Gamma$ is satisfiable iff every finite subset of $\Gamma$ is satisfiable. ($\Gamma$ satisfiable means that $\Gamma \nvDash \bot$) – 2017-01-09
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    I tried to prove the first hint (I think I can manage it), but I don't understand how to use this hint in the proof of the equivalence – 2017-01-09
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    @FlaB: A topological space space $X$ is compact if and only if every [centred](https://en.wikipedia.org/wiki/Finite_intersection_property) system of closed sets in $X$ has non-empty intersection. – 2017-01-09

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