What's the difference between open and closed sets? What's the difference between open and closed sets?
Especially with relation to topology - rigorous definitions are appreciated, but just as important is the intuition!

The set $\tau$ of open subsets of a set $X$ is an algebraic structure with $\cup$ and  $\cap$. The intersection of two sets in $\tau$ should be a set in $\tau$. And the union 
$\displaystyle\bigcup_{i\in I}\mathcal O_{i}$ should be in $\tau$ for any set of open sets $\{\mathcal O_{i}\}_{i\in I}$. Also $\emptyset,X\in\tau$. And that's all.
The set $\sigma$ of closed subsets of a set $X$ is the set 
$\{\complement_X\mathcal O \}_{\mathcal O\in\tau}$ which is the dual structure such that the union of any two closed sets is a closed set and such that $\displaystyle\bigcap_{i\in I}\mathcal F_{i}$ is closed for any set of closed sets $\{\mathcal F_{i}\}_{i\in I}$. And $\emptyset,X$ are both open and closed.
Any of those two structures defines a topology on $X$, which adds a concept of proximity to $X$. With a topology on a set $X$ it is not only possible to decide what elements thats belongs to a subset of $X$, but also which elements in $X$ that is proximate to that subset:
$x\in \overline A \Leftrightarrow \forall\mathcal O\in\tau:x\in\mathcal O\Rightarrow A\cap\mathcal O\neq\emptyset\qquad$ or dually
$\displaystyle\overline A=\bigcap_{\mathcal F\in\sigma_A}\mathcal F$, where 
$\sigma_A=\{\mathcal F\in\sigma|A\subseteq \mathcal F\}$.
The proximity is an extension of the membership to a set in a topological space $(X,\tau)$ (or dually $(X,\sigma)$) and can also be defined explicit and then define the topology it self:
Given a binary relation $\propto\subseteq X^2$ such that


*

*$\neg\exists x\in X:x\propto\emptyset$

*$x\in A\Rightarrow x\propto A$

*$x\propto A\subseteq B \Rightarrow x\propto B$

*$x\propto A\cup B\Rightarrow x\propto A \vee x\propto B$.


Then $\propto$ defines a topological proximity relation (and a topology) on X by $x\in\overline A\Leftrightarrow x\propto A$.
An other old man's thinking about topology
