In this paper, we present some characterizations of prime and maximal filters. Moreover, we introduce $\cap$-irreducible filters of an equality algebra, investigate some results about them and relations between maximal, prime, $\vee$-irreducible and $\cap$-irreducible filters in equality algebra. Also, we introduce spectrum of an equality algebra and prove that the spectrum endowed with Zariski topology is a compact T_0 topological space and maximal spectrum (as a subspace of that) is a compact T_1 topological space. Manuscript profile