As a continuation of [7], we introduce the notions of topological subalgebras, topological ideals and topological homomorphisms in topological BCI-algebras and study some related properties. In the section 3, we investigate the compactness in a TBCI-algebra $X$ and quotient TBCI-algebra $X/I$ where $I$ is a topological ideal of $X$. In the section 4, we introduce the notion of topological homomorphisms, study some properties for this notion and show that an open topological homomorphism $f$ from TBCI-algebra $X$ to TBCI-algebra $Y$ gives rise to a one-to-one correspondence between the closed c-ideals of $Y$ and the closed c-ideals of $X$ which contains $kerf$.