Regarding open subsets in topology (Probably due to my lack of experience with the subject, I see that my question is horribly written. If you are to answer, a beginner-friendly explanation of the basis of a topology and the topology generated by it is likely sufficient.)
I just started a course in Topology, and as I did with Abstract Algebra, I am having some trouble right off the bat (I lack the experience from Real Analysis, which I am doing simultaneously.)
In a task from Munkres, we are given the following problem: we are given a topological space $X$, and $A \subset X$. If for each $x \in A$ there is an open set $U$ containing $x$ such that $U \subset A$, we are to show that $A$ is open in $X$. What follows is my proposed solution:

Let $a \in A$. Let $U_a$ be the open set corresponding to $a$. Since we are given that $U_a \subset A$, it follows that $A = \cup_{a \in A}U_a$. Our desired result follows from the fact that a union of an arbitrary number of open sets is open.

This may be right, it may be inaccurate or it may be straight-out wrong. What scares me is that I do not understand the concept of an open set in Topology, which makes understanding bases, and moreover the Topology generated by a basis even harder to understand. I do understand the concept of open sets given our standard $\mathbb{R}, \mathbb{Q}$ and so on, and as such, I understand the examples provided in the book. I am asking for an intuitive explanation on how to recognize open sets in Topology given sets on the form $\{a,b,\cdots\}$ and how we extend this to the concept of the basis and the topology generated by it.
 A: Regarding intuition ...
I think closed sets are easier to grasp than open ones. A set is closed if it contains all of its limit points. What is a limit point of a set? Intuitively, a limit point $P$ of a set $S$ is a point that can be reached as the limit of a sequence of points in $S$. Or, equivalently, if you draw little tiny balls around $P$, then, no matter how small you make them, they will still include some points of $S$ (other than $P$ itself). The reasoning here is fuzzy, because we haven't said what we mean by "small", but you get thie idea.
Now take a disk shape in the plane (or any other shape enclosed by a curve). The points of the boundary curve are limit points of the shape. So, if the curve itself is part of the shape, then the shape is a closed set. This gives us the intuitive idea that a set is closed if it includes its boundary.
If you can understand closed sets, that's a good step, because (as you probably know) an open set is just one whose complement is closed.
In 2D and 3D, with any reasonable topology, and bounded sets, these sorts of notions will serve you well, I think. Things get trickier when you start considering unbounded sets, so some care is needed.
Intuition goes out the window when people make up strange topologies that bear no relationship to conventional ideas of space and distance. I would say that trying to develop intuition about exotic fabricated topologies is hopeless. As the comment above pointed out, any family of sets that satisfy a few (fairly weak) conditions can be used as the defining collection of open sets when forming a topological space, so there is no hope of being able to intuitively "recognize" open sets in this situation. 
Much of the above is pretty fuzzy half-truths, and would make your topology lecturer cringe, I expect. If so, maybe you should ask him (or her) to give you help with intuition.
