Show that ${x}in mathbb{B}(X)$ for every $xin X$

Question

Let $(X, tau)$ be a Hausdorff space, and let $mathbb{B}(X)$ be the Borel $sigma$ algebra on $X$. The question is,

Is it true that, if $xin X$, then ${x}in mathbb{B}(X)$?

The reason why I ask is because of the previous post I made; the answer shows that one can determine a Radon measure at ${x}$, but I need to verify that ${x}in mathbb{B}(X)$.

Matias Heikkilä · Answer

Since $X$ is Hausdorff ${x}$ (a singleton) is closed and $U = X setminus {x}$ is open. Since $B(X)$ is a $sigma$-algebra, it's closed under taking the complement:
$$
B(X) ni X setminus U = X setminus ( X setminus left{ xright} ) = left{ x right}
$$

Show that ${x}in mathbb{B}(X)$ for every $xin X$

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