While reading about general topology, I started looking at the metric topology. At one point, the interior of a topological space $X$ is defined as such:
$${E}^{\circ}=\{p\in E\ | \ B_p(\epsilon)\subseteq E, \epsilon>0\}$$
Here, $B_p(\epsilon)$ is a ball with center $p$ and radius $\epsilon$. I tried verifying that this met the general properties of an interior, and didn't have much trouble showing ${E}^{\circ}\subseteq E$, $(E\cap F)^{\circ}={E}^{\circ}\cap {F}^{\circ}$, and ${X}^{\circ}=X$. However, I don't see how ${{E}^{\circ}}^{\circ}={E}^{\circ}$. I see that ${{E}^{\circ}}^{\circ}\subseteq {E}^{\circ}$, but tried without success to see the other containment. Is there a reason why this follows simply from the axioms of what a metric is?
Futhermore, apparently another way to approach this topology is to define the family of open sets $\mathcal{T}$ as those which are the union of a family of balls. Again, I tried verifying this for myself. I see that for any $\mathcal{U}\subseteq\mathcal{T}$, $\cup\mathcal{U}$ is a union of sets which each may be represented as the union of a family of balls, so the whole union again may be represented as the union of a family of balls, namely those in each set in $\mathcal{U}$. Also, $X$ may be represented as the union of balls where I take one ball with center $p$ for all $p\in X$. Finally $\emptyset$ is just an empty union of balls. However, I was unable to show that for finite $\mathcal{U}\subseteq\mathcal{T}$, $\cap\mathcal{U}$ is again open. Is there some way to see that the finite intersection could be represented as a union of balls from the properties of the metric?