Sorgenfreyline proof or disproof

Question

$(\Bbb{R},\tau)$ is Sorgenfrey topological space, also known as lower limit topology, $x \in \Bbb{R}$ and $\mathscr{N}(x)$ shows all neighboorhoods of $x$. Prove or disprove $$\bigcap_{U \in \mathscr{N(x)}}\overline{U}=\{x\}$$

Answer 1

Pick $y \in \cap \bar{U}$ such that $y \neq x$. Since this topology is Haudorff, we can have the disjoint open set of $x$ and $y$ respectively, contradicts with the choice of $y$.

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