Continuous Mapping on Topology Space and Grid Convergence

In this article, based on page16-21 of R.M.Dudley. Real Analysis and Probability (Second Edition) (Dudley et al. 2008), the author adopts contradiction to construct mappings (i.e., grid) defined on directed set is proved to apply grid convergence to characterize continuous mapping on general topology space, making it possible to characterize continuous mapping on the space in a way similar to sequence convergence (convergence in metric) on metric space (which is also topology space). We have gained that continuous mapping on general topology space is equivalent to grid convergence in this article.