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Consider the binomial model $G^{d+1}(n,p)$ of the random $(d+1)$-uniformhypergraph on $n$ vertices, where each edge is present, independently of oneanother, with probability $p:\mathbb{N}\to[0,1]$. We prove that, for alllogarithmo-exponential $p\ll n^{-d+\epsilon}$, the probabilities of allelementary properties of hypergraphs converge, with particular emphasis in theranges $p(n)\sim C/n^d$ and $p(n) \sim C\log(n)/n^d$. The exposition is unifiedby constructing, for each such function $p$, the topological space of allcompletions of its almost sure theory. This space turns out to be compact,metrizable and totally disconnected, but further properties depend on the rangeof $p$. The convergence of the probabilities of elementary properties isassociated with a borelian probability measure on the space.

Spaces of completions of elementary theories and convergence laws for random hypergraphs