Let it be ($\mathrm{X}$,$\rho$) a metric space and let it be $\mathcal{S}$$\subseteq$$\mathcal{P}$($\mathrm{X}$) such that $A$$\in$$\mathcal{S}$ if and only if there is an open $\mathrm{U}$$\subseteq$$\mathrm{X}$ and for every $n$$\in$$\mathbb{N}$, there is a $E_n$$\subseteq$$\mathrm{X}$ such that $int(cl(E_n))=$$\varnothing$ in a such way that $A$$\bigtriangleup$$\mathrm{U}=$$\bigcup\{E_n:n\in\mathbb{N}\}$. Prove that $\mathcal{S}$ is a $\sigma$-algebra of subsets of $\mathrm{X}$.
I've already prove that $\mathcal{S}\neq\varnothing$ and that $\mathrm{X}\in\mathcal{S}$ but i don't have any idea of how to proof that $\mathcal{S}$ is closed under countable unions and that $\mathcal{S}$ is closed under set difference.
I know that i should use topological properties and set theorical operations but i'm not very good at that.
This is one of the classic $\sigma$-algebras (like the Borels sets). A set $E$ such that $\operatorname{int}(\operatorname{cl}(E)) = \emptyset$ is a "nowhere dense set". A countable union of nowhere dense sets is " a set of the first category" or a meagre/meager set. Intuitively, such sets are "small": in a complete metric space no open set can be of the first category. The $\sigma$-algebra you describe are those that are "almost open" (they differ by an open set by a meagre set. These sets have the property of Baire. Look in Oxtoby's book (Measure and Category, chapter 4) for a proof of being a $\sigma$-algebra, e.g. It's in fact the $\sigma$-algebra generated by the collection of sets that are open or meagre.