Given an abstract logic $\mathscr{L = L}(Q^i)_{i \in I}$ generated by a set of quantifiers $Q^i$, one can construct for each type $\tau$ a topological space $S_\tau$ exactly as one constructs the Stone space for $\tau$ in first-order logic. Letting $T$ be an arbitrary directed set of types, the set $S_T = \{(S_\tau, \pi^\tau_\sigma)\mid\sigma, \tau \in T, \sigma \subset \tau\}$ is an inverse topological system whose bonding mappings $\pi^\tau_\sigma$ are naturally determined by the reduct operation on structures. We relate the compactness of $\mathscr{L}$ to the topological properties of $S_T$. For example, if $I$ is countable then $\mathscr{L}$ is compact iff for every $\tau$ each clopen subset of $S_\tau$ is of finite type and $S_\tau$ is homeomorphic to $\underset{lim}S_T$, where $T$ is the set of finite subtypes of $\tau$. We finally apply our results to concrete logics.