Real numbers having disjoint neighborhood How does one prove that each point in real line has disjoint neighborhoods?
HINT: Do you mean that any two points on the real line have disjoint neighborhoods? If so, consider that for any real numbers $x$ and $y$ with $x\ne y$, the average $\frac12(x+y)$ is between $x$ and $y$. (Exactly halfway between them, in fact, though you don’t need that.)

If you really meant that there is a pairwise disjoint collection $\{V_x:x\in\Bbb R\}$ of open sets such that $x\in V_x$ for each $x\in\Bbb R$, that’s simply false: every pairwise disjoint family of non-empty open sets in $\Bbb R$ is countable, because each of them must contain a different rational number, and there are only countably many rational numbers.
Since $\mathbb{R}$ is Hausdorff, distinct points in $\mathbb{R}$ have disjoint neighbourhoods.