On thinking that planarity is nothing but topology?

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I've found the following quote on Harary's Graph Theory:

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And I'd like to know what it means. I know about the Kuratwoski theorem, which states that a graph is planar if no subgraph of it is isomorphic to $K_{3,3}$ or $K_{5}$ but I have no idea on what is the meaning of the rest of the quote. At the moment, I am having a course on graph theory but I still don't know much what topology is.

  • In the quote, he seems to be saying that thinking that planarity is nothing but topology is a problem. Why is that?