Definition of topological space The definition of a topological space is a set with a collection of subsets (the topology) satisfying various conditions. A metric topology is given as the set of open subsets with respect to the metric. But if I take an arbitrary topology for a metric space, will this set coincide with the metric topology? 
I'm trying to justify why we call the elements of a topology "open". If my above question is true, then at least in a metric space, the set of open sets is equivalent to the topology of the metric space. 
So am I right in thinking that when we remove the metric, we are generalising this equivalence by defining the open sets as those that satisfy the conditions of a topology?
 A: Except for the trivial case of a metric space with only one element, there is always at least one topology on a metric space that is not the same as the metric topology, namely the discrete topology in which only the empty set and the whole space are open. An example of a very interesting topology on $\Bbb{R}$ that is not the metric topology is the lower limit topology.
As the answer to your first question is "no", you may want to rethink your second question.
A: Suppose you've got a set, $X$. If you equip $X$ with a metric $d$, now the pair $(X,d)$ is a metric space.
This metric generates a topology on $X$. You consider the collection $\mathcal{B}$ of sets of the form
$$
B(x,r):= \{y \in X:d(x,y) < r \}.
$$
Now the collection $\mathcal{B}$ is not a topology in and of itself, but lets you build one by taking arbitrary unions and finite intersections of sets from $\mathcal{B}$. The resulting collection $\mathcal{T}$ satisfies the conditions needed for a collection to be considered a topology on $X$. This is exactly the metric topology on $X$ with respect to $d$. So in this setting, a subset of $X$ will be open (is in $\mathcal{T}$) if and only if it is open with respect to the metric (that you can fit a ball of some radius around each point in the set).
But you could just as easily muster up a collection of subsets $\mathcal{T}'$ of $X$ that don't spawn from a metric on $X$, yet still satisfy the conditions to be a topology. The open sets in $(X,d, \mathcal{T})$ will be in general very different from those in $(X, \mathcal{T}')$.
You can build a topology from scratch by declaring, "I want these sets to be open." Then you do what's necessary to make sure everything satisfies the conditions to have a topology. This is the concept of a basis for a topology. So in this setting the idea of "open" in topological spaces is really just an abstraction of the "open" you're used to in metric spaces. 
