Let $X$ be a complex space, $\{ X_{i} \}$ an increasing sequence of open subspaces of $X$, such that $X = \bigcup X_{i}$ and $\mathfrak{F}$ a coherent analytic sheaf on $X$. If the restrictions $\mathfrak{F}|X_{i}$ have certain cohomological properties, then we prove that similar properties hold for $\mathfrak{F}$ on the whole space $X$. For instance, if $H^{j}(X_{i}, \mathfrak{F}) = 0$ for every $j \ge 1$, then we prove that either $H^{j}(X, \mathfrak{F}) = 0$ for every $j \ge 1$, or $H^{1}(X, \mathfrak{F})$ has no Hausdorff topology.