Is there a good visual aid or picture to help understand openness and closedness? I'm struggling to grasp the idea of open, closed, clopen and not open and not closed sets in a more formal approach like how it's described in a math analysis class. Is there a good picture somewhere that can help describe it? Or perhaps you know how to describe it in a good way that differs from other textbooks?
 A: Closedness is checking the complement for openness.
If A has no boundary points , A is open.
If A has all the boundary points , A is closed.
A: A set $A$ is said to be open if and only if $\operatorname{int} A=A$ and a set $A$ is said to closed if and only if the closure $\operatorname{cl}(A)$ equals $A$.
A: If you're in a metric space $X$, then a subset $S\subseteq X$ is closed if and only if, whenever $(x_n)_{n\geq 1}\subseteq S$ is a convergent sequence with limit $x=\lim_{n\to\infty} x_n$, then $x\in S$. So a closed set is a set which you cannot escape using convergent sequences. 
A: The following work in any metric space $X$.
Approach 1.
A set $B$ is open iff for all $b \in B$ there an $\epsilon$ of room. That is, a set $B$ is open iff for all $b \in B$ we can find $\epsilon > 0$ such that the ball of radius $\epsilon$ centered at $b$ is a subset of $B$. Thus an open set is "roomy" or "spacious." A set $B$ is closed iff for all $b \in B^c$ there an $\epsilon$ of room. That is, a set $B$ is closed iff for all $b \in B^c$ we can find $\epsilon > 0$ such that the ball of radius $\epsilon$ centered at $b$ is a subset of $B^c$. Thus a set is closed iff it has a "roomy" or "spacious" complement.
Approach 2.
A point $x \in X$ is a boundary point of $B$ (but not necessarily an element) if every (non-empty) ball centered at $x$ overlaps somewhat with $B$, and also overlaps somewhat with $B^c$. A set is open iff every boundary point of the set fails to be an element of the set. A set is closed iff every boundary point of the set succeeds in being an element of the set.
