Topological graphs I am studying graph theory from perspective of topology. Wikipedia says that topological graph is a topological space constructed from regular graph by replacing vertices with points and edges with intervals. I am not sure what are intervals. I suppose that these are some kind of topological space and I've read that they are called manifolds. I've read also that topological homeomorphism is equivalent to topological graph isomorphism. And I kinda understand, but can you, please, clarify, can edges of such graph intersect with each other. Because I feel that two intervals that intersect is not the same topological space. So, how can I understand this? Thank you in advance!
Edit:
And another question. Am I mistaken when I said that homemorphism of surfaces is equivalent to graph homeomorphism? So, can we from isomorphism of two topological graphs (there I mean homeomorphism) assert that two surfaces are equivalent?
 A: It is common in topology to take an archetypal topological space --- for example THE interval $[0,1]$ --- and then borrow that terminology, using an indefinite adjective, to represent any homeomorphic topological space. Thus we have:

An interval is a topological space homemorphic to the interval $[0,1]$.

There are other common examples of this, for instance circles and spheres.
Your post continues with a bunch of other questions which get quickly get out of focus, so I will close by addressing only the most focussed of them. A topological graph does indeed contain a specified collection of intervals (meaning, according to the definition above, subspaces each of which is homeomorphic to the interval $[0,1]$). These specified intervals are themselves still called the edges of the graph. It is indeed possible for two edges $E_1,E_2$ of a graph to intersect each other, but that is only possible if the intersection $E_1 \cap E_2$ consists of a single point $p$, and that point $p$ must be an endpoint of the interval $E_1$, and it must also be an endpoint of the interval $E_2$.
