Too long for the comments:
You are confusing a circle with a disc. A circle in a topological context is (homeomorphic to) the boundary of a disc. A circle has empty interior, in part because it is a boundary. One can of course speak about the region of $\Bbb R^2$ that is enclosed by the circle as, in some sense, its "interior". However, when you embed a circle in say $\Bbb R^3$, this notion of interior no longer makes sense, while $S^1 \subseteq \Bbb R^3$ still has empty (topological) interior.
Furthermore, neither a circle, nor a disc is open and closed in the standard topology on $\Bbb R^2$. The circle is closed, as is the disc. You should try and convince yourself that they are not open.
There is no problem with using a metric to define a topology. And you don't need a topology to define a metric space, you just need a metric. Surely you don't need to talk about open sets to construct a function $d: X \times X \to \Bbb R$?
A topological space is not really "more fundamental," it just happens that every metric space has a topology that is compatible with the metric in some sense. This is what we should want to happen, since topological spaces are generalizations of metric spaces.
You should also bear in mind that the definition of what an open set is depends on what sets you choose to be open. There are many different ways of choosing those open sets, many of which are compatible with various other mathematical structures. In addition to metric spaces, if a set has a linear order, then we can define in a particular way, a topology on that set using the linear order. When we look at algebraic objects, like groups, it might be possible to find a topology on the underlying set such that the group operation the map $g \mapsto g^{-1}$ are continuous. (This is called a topological group.)
In fact, $\Bbb R$ with the standard topology is an example of all of these. It is a metric space, linearly ordered, and is a group under addition. Just to be clear, the standard topology on $\Bbb R$ is generated by the Euclidean metric, but can also be generated by the linear order $<$, and makes addition continuous. This single, easily defined topology allows us to use three (actually more) extremely useful objects when discussing the topology of $\Bbb R$.
Remember that a topology on a point-set $X$ is just a collection of subsets of $X$ such that the collection satisfies certain properties. Look at the abstract definition of a topology. It only requires three axioms:
$\emptyset$ and $X$ are in the topology.
That the collection is closed under arbitrarily large unions.
That the collection is closed under finite intersections.
So that it is easier to talk about members of the collection, we call them open sets.
Observe that nowhere in the definition of what a topology is do we give any characterizations about the open sets themselves. This is somewhat analogous to how the axioms of a group make no mentions of what the group elements are, only how they interact.
The advantage to this flexibility is that it is easy to put different kinds of topologies on an arbitrary point-set. The down-side is that often we have too many choices and the easy ones (the discrete and trivial topologies) are rarely useful or what we want (but sometimes they are!).
Let's say you have a point-set $X$ and you want to put a topology on it. Chances are that you have some additional structure on $X$ and you want to use topological tools to study it. In that case, whatever topology you pick needs to be compatible with that structure in some sense that will vary a lot between structures.
The prototypical example is the metric topology. Suppose $(X,d)$ is a metric space. From analysis we have a definition of continuity on $X$, but we also have a different definition for arbitrary topological spaces:
Analytic Continuity: A function $f: X \to Y$ is continuous at $x \in X$ if for all $\epsilon > 0$, there exists a $\delta > 0$ such that for all $y \in X$, if $d_X(x,y) < \delta$, then $d_Y(f(x),f(y)) < \epsilon$.
Topological Continuity: A function $f: X \to Y$ is continuous if for all open sets $U \subseteq Y$, the preimage $f^{-1}(U)$ is open in $X$.
Whatever choice we make of topology on $X$, we want these two definitions to be equivalent, so that we can combine the analytic and topological machinery.
The take-away of all this is that when you just have a point-set and nothing else, it's not clear what the open sets should be or why you would even need these open set doohickeys. In fact, when we make purely abstract point-set topology arguments, it doesn't matter what an open set is, since we are reasoning about them abstractly. Once we start looking at (slightly) more concrete mathematical structures, like say metric spaces or groups, we can no longer reason about open sets abstractly, since there are certain compatibility properties we would like them to satisfy. Now we need to single out a particular collection of subsets that we would like to call the open sets and show that is forms a topology.
The other sensible choice of what a continuous function is what we call an open function:
- Open Function: A function $f: X \to Y$ is called open if for all open sets $U \subseteq X$, the image $f(U)$ is open in $Y$.
So why don't we call these functions continuous?
The short (unhelpful) answer is because the preimage definition is what works. To give a precise mathematical reason is not easy, because essentially our choice was based on what got us what we wanted. Any good definition of continuity ought to formalize our intuitive notions of continuity and the preimage definition does that better than the image definition.
It is also worth pointing out that preimages play much nicer with set operations than do images. To wit:
$f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B)$, whereas $f(A \cap B) \subseteq f(A) \cap f(B)$ and
$f^{-1}(A \setminus B) = f^{-1}(A) \setminus f^{-1}(B)$, whereas $f(A \setminus B) \supseteq f(A) \setminus f(B)$.
And since whichever one choose in the definition will be used a lot with sets and set operations, so from a practical standpoint, we kind of want the preimage to be the right definition.
Maybe an analogy will help. Supposing you start with an understanding of Real Analysis, you can generalize this understanding to Metric Spaces:
In Real Analysis, you have this fixed notion of distance between two points in $\mathbb{R}$ defined as $|x-y|$. Then using this definition you build your collections of open sets that you use to study $\mathbb{R}$. These sets look like open intervals $(x-r, x+r)$ centered at $x$ with radius $r$, and using these open sets you can study the space (continuity, connectedness, derivatives, etc.)
Then you jump to into a general a Metric Space and are told "Use whatever definition of distance that you like (as long as it follows a few rules to keep things sensible) and build your open sets with that!" Then you just proceed to do the same thing you did in Real Analysis only you are using open balls that look like $B(x,r)$, centered at $x$ with radius $r$.
Now we can see a related jump from understanding Metric Spaces to understanding General Topology:
In Metric Spaces, you have this fixed notion of distance between two points in your space $X$ defined as $d(x,y)$. Then using this definition you build your collections of open sets that you use to study $X$. These sets look like open balls $B(x,r)$ centered at $x$ with radius $r$, and using these open sets you can study the space (continuity, connectedness, derivatives, etc.)
Then you jump to into General Topology and are told "You don't need to bother building the collection of open sets at all. Instead, I'm just going to hand you this collection of open sets! (and these open sets will follow a few rules to keep things sensible)" Then you just proceed to do the same thing you did with Metric Spaces only you are using these open sets that you were handed to you, and there might not be a easy way to describe what they look like.
Best Answer
The heart of it is that, while your intuition is that "continuity" is about distances - you first learn continuity in terms of $\epsilon-\delta$ proofs - it turns out that continuity is really only about open sets. Whether a map $f:X\to Y$ between metric spaces is continuous is entirely determined by the open sets of $X$ and $Y$. If you have two different metrics for $X$ which determine the same open sets, the set of continuous functions on $X$ are the same.
Since topology is the study of continuity, it makes sense to only care about open sets, then, not metrics.
The original definitions for "topology" had more rules about the open sets that made topologies "more like" metric spaces. But as mathematicians started playing with these things, they found that it made sense to ask questions about continuity in the cases where these rules were broken, too. So the definition was broadened to give the widest meaning.
A very basic example might be left-continuity. A function $f:\mathbb R\to Y$ with $Y$ a metric space is called left-continuous if $\lim_{x\to a-} f(x)=f(a)$ for all $a\in\mathbb R$.
It turns out, there is a topology on the real line, call it $\tau_{L}$, under which a function $f$ is left-continuous if and only if it is (topologically) continuous as a function from $(\mathbb R,\tau_L)\to Y$. (Note: $\tau_{L}$ has a basis the intervals of the form $(x,y]$.) Now, $\tau_{L}$ is a point-set topology, but it is not one that comes from a metric. So the notion of "left-continuity" is an example of an idea that fits the point-set definition of continuity, but does not fit the $\epsilon-\delta$ notion of continuity - you essentially need to redefine it.
The fact that left-continuity and right-continuity together is the same as "normal" continuity is a statement about the three different topologies. Somehow, the usual real line topology is a combination of these two other topologies.
It turns out there is something really deep going on here. Somehow, the open neighborhoods of a point in a topology contains a lot of information - what we call "local information" - about behavior of "nice" functions at that point.