I see a lot of literature( example) that says: consider graph $G$ as a one-dimensional topological space. We know that the definition of a topological space via open sets can be defined as follows:(https://en.wikipedia.org/wiki/Topological_space)
A topological space is an ordered pair $(X,τ)$, where $X$ is a set and $τ$ is a collection of subsets of $X$, satisfying the following axioms:
1.The empty set and $X$ itself belong to $τ$.
2.Any arbitrary (finite or infinite) union of members of $τ$ belongs to $τ$.
3.The intersection of any finite number of members of $τ$ belongs to $τ$.
The elements of $τ$ are called open sets and the collection $τ$ >is called a topology on $X$.
I'm wondering what is an open set for the graph if I follow the strict definition of this topological space. Does this space with one-dimensional dimensions refer to the drawing of the graph on surface((maybe a sphere, a torus) rather than the abstract graph itself?
If we think about drawing, maybe a graph can be drawn in different ways. Do they all topological Spaces? If there are crossings (may not the vertices) between the edges of a graph(for example： $K_{3,3}$ when it was drawn on the sphere), is it still a topological space?
There were some questions about graph topologies in the forum, but my doubts remain.