Metric spaces:
Neighborhood of a point $a$ is a Set of point $N$, such that $\exists\delta>0:B_\delta(a)\subset N$ ($B_r(x)$ = open ball at x of radius r)
Definition of open set: "A subset $O$ of a metric space is said to be open if it is a neighborhood to each of its points"
(It is now easy to go from this to the metrisable topological spaces)
Continuity:
($a$ is a point, $M$ and $N$ are neighborhoods (of the form $\forall M\exists N$), $M$ is a neigh. to $f(a)$, $N$ of $a$)
$\epsilon-\delta \iff f(B_\delta(a))\subset B_\epsilon(f(a))\iff B_\delta(a)\subset f^{-1}(B_\epsilon(a))\iff f(N)\subset M\iff N\subset f^{-1}(M)\iff f^{-1}(M)\text{ is a neigh. of }a$
This shows the "list" going from the well known $\forall\epsilon>0\exists\delta>0:|x-a|<\delta\implies|f(x)-f(a)|<\epsilon$
requiring only a leap of faith from $|\text{_}|$ to metrics (the ball definition) then it's straight on to the more topological $\forall N_{f(a)}, f^{-1}(N_{f(a)})\text{ is a neigh. of }a$
Topological space:
Given a topological space $(X,J)$ a set $N\subset X$ is a neighborhood of a point $a\in X$ if $N$ contains an open set that contains $a$
Definition of open set: "Thee members of $J$ are called 'open sets'"
Question
The leap of faith ("Trust me, we're not just calling these 'open sets' rather than 'Thingymajig', think of them as open sets" and "Yeah, these are 'neighborhoods' we give them this name because they are like neighborhoods from metric spaces') required is really quite large.
I know it cannot be proved (there's no distance thus no open balls for the general topology) but I'd like some more discussion about accepting these definitions. From what I can tell.
"If we call an open set this, and a neighborhood that, later when we deal with distance and metrics, we can define open sets and neigh. using the metric, and show it has these properties, the that leads to continuity as you think of it"
So it's sort of going "from topology, to metric spaces" in terms of definitions and showing properties, however every book on the subject goes the other way. It builds on metric spaces then suddenly "we call these open sets!" I'd like to be a bit more confident and happy with the transition, recommended reading, examples, anything would be great.
When we say a metric topological space, we mean that opens sets in this topology is induced by the metric. So in this case definition for open sets is unique.
As for difference between open set and neighborhood, remark that each open set is a neighborhood for points inside it, and that each neighborhood contains an open neighborhood. So finally they are so "equivalent"(Of course not really, since a neighborhood is not always open) that usually we can interchange their names in most cases without making real mistakes