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We provide a residual-based a posteriori error estimate for a space-time formulation and the corresponding efficiently computable bound for the<jats:italic>certification<\/jats:italic>of the method. We introduce a Petrov-Galerkin finite element discretization of the continuous space-time problem and use it as our reference in a posteriori error control. The Petrov-Galerkin discretization is further approximated by the Crank-Nicolson time-marching problem. It allows to use a POD-Greedy approach to construct the reduced-basis spaces of small dimensions and to apply the Empirical Interpolation Method (EIM) to guarantee the efficient<jats:italic>offline-online<\/jats:italic>computational procedure. In our approach, we compute the reduced basis solution in a time-marching framework while the RB approximation error in a space-time norm is controlled by our computable bound. 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