General topology for physicists In the definition of continuity on sets, it is said that one has to have a topology constructed for that set, id est, a topological space. 
If we now have two such sets and a map betwen them we say that this map is continuous if its pre-image of one element of the topology in the target set lands in some element of the topology of the domain set. 
So, it has to preserve topological structure. Why is continuity defined in this way? What is the intuition? And I know about standard topology on $\mathbb{R}^n$ so don't say that. It's not likely that this is the only reason general continuity would be defined in such a way. Also, I am aware of the dangers of this question being proclaimed to be for mathematicians. I believe that for this special case of problems I, as a physicist, need to post my question here.
 A: In bare, simple, but probably not too satisfying terms, it is because neighborhood, as a topological notion, is the definition of closeness and continuity, intuitively, means the conservation of closeness and neighborhood under the transformation in question.
But why is such a seemingly abstract definition related to closeness? Concrete answers are well written in Yuggib's and Mike Stone's answers. Hopefully Mike's answer shows you that a special case of the definition is Weierstrass's original $\epsilon-\delta$ one, which should be fairly clear.
If you want to ask why further to Weierstrass's definition, then it becomes a matter of opinion - you simply have to sit down with your pencils and paper and sketch and see that the definition captures the intuitive idea of keeping neighbors near to one another. I think neither theoretical physicists nor mathematicians should be frightened of doing this "pedestrian" activity - definitions ultimately are about capturing intuition.
Your question is a good one because a deeper answer illustrates well the genius that is hidden in the definitions of mathematics. Mathematics texts seldom convey the sheer work that great mathematicians had to do to get to the definitions that make things seem to work so smoothly for us today. I still wonder at the slickness of topology, in particular, in capturing deep properties so simply. The little cogs and gears of topology fit together so effortlessly and beautifully: one has to read the history a little to understand that definitions don't simple appear by magic. The formulation of definitions is very much an experimental science: one tries an intuition, then, through rigorous theorems and inferences from the definition, one sees how well one can reproduce other intuitions. I'm sure that mostly this process ends in bitter disappointment, the waste paper basket (or more recently /dev/null and other embodiments of the dead bit bucket at the end of the Universe) but we never see these dead end attempts, of course!
In this particular case, the genius here belongs mostly to the Russian point set topology school of Pavel Alexandrov and Pavel Urysohn. The evolution of the definition that troubles you happenned in the late 1920s. You probably ken Alexandrov's name from the elegant One Point Compactification Procedure, of which the Riemann sphere from $\mathbb{C}$ can be thought of as an example.
Beginning with Weierstrass's definition (and this is my speculation trying to construct a plausible story for what might have gone on in the heads of the Russian scholars), someone recognized that the essence of the idea of continuity was not the inequalities, but the notion of neighborhood and closeness. So why not make these latter the primitive notions? One could then broaden the notion of continuity beyond $\mathbb{R}^N$. That's a fairly concrete objective for a mathematics project, but it wouldn't have been easy to see a way forward. At some point the idea of membership of a class of sets defining prototypical neighborhood and closeness came into view. A mid point was the notion of metric space That's where topology comes from. Later, Urysohn did some "back filling" to shore the notion up by arriving at the Urysohn Metrization Theorem; this must have been a truly stunning victory for the school, because it shows how closely the more general notion of topology co-incides with the more specialized ideas of metric space and Weierstrass continuity. For a topological space not to be metrizable, it must lack very fundamental properties. 
A physicist would think of a non metrizable space as pretty pathological. Urysohn metrization assures us that topology is just as good as Weierstrass continuity, and easier to work with, in all the cases that are likely to interests most physicists!
One is also tempted to speculate that the spur to abstract the notion of continuity got some thrust from Emily Noether's "Begriffliche Mathematik" philosophy that was growing its legs at the time. Certainly, there was fairly tight contact between Alexandrof's school and Hilbert's.
Another reference that might work for you would be to read the Chapter "Neighborhoods and Neighborhood Spaces" in Bert Mendelson's "Introduction to Topology"(section 3.3). I think I really appreciated the achievements of the point set topology school by reproducing the simple proofs for myself in this section. I think this book is a must before tackling something like Munkres's Topology. Somehow I've never liked the latter. In keeping with your question title, I do like (although it is old), John Kelley's "General Topology".
A lovely, relevant intuition that I've always treasured from Mendelson's book is his view of the triangle inequality: it is expressing the transitivity of closeness and assures us that if $A$ is near to $B$ and $B$ near to $C$, then $A$ must be "quite near" to $C$, by dint of the bound!
A: An analogous, and more intuitive, definition of continuity is by means of neighbourhoods of points. A neighbourhood of a point $x\in X$ is a set that contains an open set containing $x$. So if $z\in V$, where $V$ is a neighbourhood of $x\in X$, we can picture that $z$ is "$V$-close" to $x$. A function $f: X \to Y$ is continuous at $x$ if for any neighbourhood $V′$ of $f(x)\in Y$, there exists a neighbourhood $V$ of $x\in X$ such that $z\in V$ implies $f(z)\in V'$. This definition is quite pictorial, it means that if $f$ is continuous at $x$, then $f(z)$ can be as close to $f(x)$ as we want (it is $V'$-close, with $V'$ arbitrary), as long as as $z$ is sufficiently close to $x$, i.e. as long as $z$ is $V$-close (and we are assured that at least one such $V$-close $z$ exists by definition of continuity).
A function is then continuous if it is continuous at every point. And for continuous functions, the definition above is easily proved to be equivalent to the fact that the preimage of  any open set is open. 
A: It's defined that way so as to coincide with the notion of continuity on ${\mathbb R}^n$.  If  $f:{\mathbb R}^n\to {\mathbb R}$, the   statement that for all $\epsilon>0$ and for all $x_0$ there exists a $\delta>0$ (depending on $x_0$) such that $|x-x_0|<\delta$ implies that $|f(x)-f(x_0)|<\epsilon$   is equivalent to the statement  that the inverse image of an open set is itself open. This is becuase we know that if $f(x_0)$ is in the open set in the target space it has a  $\epsilon$-ball surrounding it that is in the target open set, and then  the open $\delta$-ball around $x_0$ maps into the $\epsilon$-ball.    
