TITLE: what exactly is an open set?
QUESTION [6 upvotes]: Many, infact  all the books on topology I have come across define open sets in the following way:

"A set $A$ is said to be open if by moving in small amounts in any direction about any point 
  we land up at a point which belongs to the same set."

Is it so that an open set is always a collection of points only? OR does there exist a general definition of open sets, without taking points into consideration?

REPLY [4 votes]: The "small amounts in any direction" idea doesn't have any direct translation to topology, but another similar idea has an exact definition in topology:  here, open sets are intuitively those sets which surround all the points they contain.
The justification of this is as follows:
Start with any topological space and two subsets $A$ and $B$ inside that space.  Now in a plain old set, either $A$ and $B$ intersect or they do not. However, in a topological space, we can formalize the idea that $A$ and $B$ 'touch', if not actually intersect.  Say that $A$ and $B$ 'touch' if every open set containing $A$ intersects $B$ or every open set containing $B$ intersects $A$ [for future reference: this happens iff 'the closure' of the two sets intersect in the usual sense]. 
For example, on the real line, $A = [0,1)$ 'touches' $B = [1,2]$.  Why?  Because any open interval containing $B$ will spill over enough to detect an intersection with the nearby set $A$.
Back to the idea of open sets as surrounding sets.  By definition, any point inside an open set $U$ automatically does not 'touch' anything outside that set because by definition the open set $U$ is proof that it doesn't!  
This gives a (admittedly rather vague) sense that a point in an open set is spatially separated from the points outside that open set.