Let $X = \mathbb{R} \setminus {0} \{0 \} \cup \{ a,b}$a,b\}$. Hence$X$is the real line sans the origin with two points$a\neq b$, both not in$\mathbb{R}$, thrown in. The topology is generated by the open intervals in$\mathbb{R} \setminus {0}$along with sets of the form$(u,0)\cup {a} \cup (0,v)$and$(u,0)\cup {b} \cup (0,v)$, where$u < 0 < v$.$X$is not Hausdorff because$a$and$b$cannot be separated by disjoint open sets. Every sequence that converges to$a$also converges to$b$. Eg.$1/n \to a$and$1/n \to b$. 1 Let$X = \mathbb{R} \setminus {0} \cup { a,b}$. Hence$X$is the real line sans the origin with two points$a\neq b$, both not in$\mathbb{R}$, thrown in. The topology is generated by the open intervals in$\mathbb{R} \setminus {0}$along with sets of the form$(u,0)\cup {a} \cup (0,v)$and$(u,0)\cup {b} \cup (0,v)$, where$u < 0 < v$.$X$is not Hausdorff because$a$and$b$cannot be separated by disjoint open sets. Every sequence that converges to$a$also converges to$b$. Eg.$1/n \to a$and$1/n \to b\$.