Level-set method is a flexible and efficient tool for structural topology optimization. Here, a technique based on space-time formulation in the context of an integral-free meshfree method in a Eulerian approach is presented for interface taking using the level-set method. We use radial basis functions for function approximation in the space-time domain. The technique does not need to march in time to obtain interface development. It obtains instantly at all defined times by solving sparse and symmetric linear equations. In this paper, the time-slab concept is applied to reduce the computational cost of the procedure. The proposed technique provides the possibility of applying effective adaptive refinement techniques in domain discretization for the transient problem. A vectorized formulation of the proposed approach is illustrated in this paper which makes it efficient in calculation and readable for programming. Two problems with analytical solution are solved to evaluate the accuracy of the presented technique.