Open Sets Definitions I took a real analysis course in my second semester. We studied about metric spaces from baby Rudin. In that it was clearly defined that w.r.t Metric Spaces that a set is open if for every point inside it we can find a neighborhood of that point which lies completely inside that set.
Now in my fourth semester, I am taking a course in Introductory Topology. We are following Munkres. In the book, its given that if a subset  $U$ belongs to a topology of a set $X$, then $U$ is open in $X$.
I am really confused between the two as they are being used interchangeably. Can anybody please explain whether are the both definitions equivalent or not?
 A: In a metric space, a set is open if it is a member of the (implicit) topology consisting of all sets described via the neighborhoods as you do. The topology (as a set of open sets) is not mentioned explicitly, but it is there.
The equivalent notion in an arbitrary topological space is that of a topology generated by a basis: a basis is a set of elements of the topology such that given any element $U$ of the topology and any $x\in U$, there exists a $B$ in the basis such that $x\in B\subseteq U$. The given basis for a metric space is the set of all open balls.
A: Studying metric spaces one quickly realizes the fundamental role played by the open sets (as defined in the original post).
This is attested by results showing that several other important notions may be phrased in terms of open sets alone (not involving the metric).
For example: a function is continuous iff the inverse image of every open set is open; a sequence converges to a point $a$ iff every open set containing $a$ eventually contains all points of the sequence.
This leads one to suspect that the metric is disposable, that is, if one knows which sets are open and nothing else, one may develop large parts of the theory unhindered.
This is the idea behind the definition of a Topology, where, ignoring any metric, one postulates a collection of sets to call the open sets.
