In this paper we introduce stable topology and $F$-topology on the set of all prime filters of a BL-algebra $A$ and show that the set of all prime filters of $A$, namely Spec($A$) with the stable topology is a compact space but not $T_0$. Then by means of stable topology, we define and study pure filters of a BL-algebra $A$ and obtain a one to one correspondence between pure filters of $A$ and closed subsets of Max($A$), the set of all maximal filters of $A$, as a subspace of Spec($A$). We also show that for any filter $F$ of BL-algebra $A$ if $\sigma(F)=F$ then $U(F)$ is stable and $F$ is a pure filter of $A$, where $\sigma(F)=\{a\in A| y\wedge z=0$ for some $z\in F$ and $y\in a^\perp\}$ and $U(F)=\{P\in $ Spec($A$) $\vert F\nsubseteq P\}$.

BL-algebra, prime filters, maximal filters, pure filters, stable topology, F-topology

03G25, 06F99, 08A72