I'm having trouble understanding the notion of an open set when applied to a space without a metric defined on it. I have read that all metric spaces are naturally a topological space, but the converse is not true. Topological spaces definitions I have read use the idea of open sets, and I can't understand this abstract idea of openness or closedness of a set without having a notion of distance.
I can understand the notion of open and closed sets in a metric space from the definitions I have read using the idea of distance and open balls. So for example given a metric space $M$ with the metric $d$, we can say a set $U \subset M$ is open if $\forall x \in U, \exists B(x,r) \subset U \dots$ which says that we can choose any point within the set $U$ and there will always be some sufficiently small distance $r$ that we can move in any direction to another point $y$ that is also contained within $U$. This has an intuitive conceptual meaning in my head regardless of space $M$ or metric $d$.
Now take the example of a space $F$, a fruit bowl with $3$ apples, $3$ oranges and $2$ bananas. There is no metric defined on the space $F$ to determine a distance between its elements, the fruit. Can we define an open set on this space?
In order to be a topological space we need to be able to define open sets right?