uniqeness of limit in topological spaces let $(X,\tau)$ be a topological space such that for every $x\in X, \bigcap_{x\in U\in \tau}U= \{x\}$. Does that imply that every convergent sequence in $X$ has a unique limit?
 A: No. It's exactly equivalent to $X$ being $T_1$, so in particular it's true if $X$ is an infinite space with the cofinite topology. Now show that any sequence of distinct points in such an $X$ converges to every point of $X$.
A: Endow $X=\mathbb{R}$ (or any infinite set) with the cofinite topology.
For each $t \ne x$ consider the open set $V_t=\mathbb{R}-\{t\}$. It follows that:
$$ \{x\}=\bigcap_{t\ne x}{V_t}$$
Therefore this space has your desired property.
In this topology almost all sequences converges to every points. 
For example let $x_n= n$. Take $a\in \mathbb{A}$ We prove that $x_n \to a$ in this topology. 
Let $V$ be an open set containing $a$, by definition of cofinite topology this open set has the form $V=\mathbb{R}-\{b_1,...,b_n \}$ (with $b_i\ne a$). Choose $N$ such that $N> b_i$ for each $i=1...n$. Therefore $x_n \in V$ if $n \ge N$, in other words $x_n \to a$. for each $a\in \mathbb{R}$. 
A: Let $X$ denote the line with two zeros. It satisfies the above property. However, any sequence converging to $0$ on the standard line converges two both zeros on $X$. Hence, the answer in no.
