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 $1/n \to b$.
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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 $1/n \to b$. |
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