The Open Set $X-\lbrace x \rbrace$

Question

I am task with proving the following: if $x \in X$ then $X- \lbrace x \rbrace $ is an open set

I kind of have an idea but I am unsure about it and how to express it.

I was thinking about using the Hausdorff property i.e. $\forall y \neq x \in X$ $\exists U_y $ where $U_y$ is open and where $y\in U_y$ but $x\notin U_y$ and then take the union of these sets $U_y$

Answer 1

Let $y\in X-\lbrace x\rbrace$. Then $y\neq x$ and thus there is an open $U_{y}$ such that $y\in U_{y}$ and $x\notin U_{y}$. Thus $U_y\subset X-\lbrace x\rbrace $. Thus it's open because it contains an open neighbour for every element.