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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?

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There were some questions about graph topologies in the forum, but my doubts remain.

licheng
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  • This is a decent question. Does graph isomorphism correspond to topological homeomorphism? – Randall Oct 08 '21 at 03:24
  • @Randall It should not be exactly the same. Graph isomorphism is for abstract graphs. But topological isomorphism has its own definition, for the homeomorphism of two drawings. – licheng Oct 08 '21 at 04:05
  • Do you know about CW complexes? Or quotient topology? – Moishe Kohan Oct 08 '21 at 04:09
  • @MoisheKohan My main interest is graph theory, and topology is self-taught a little but far from enough, what do you mean a graph is a 1-dimensional CW complex in which the 0-cells are the vertices and the 1-cells are the edges. The endpoints of each edge are identified with the vertices adjacent to it. (see https://en.wikipedia.org/wiki/CW_complex） I'd love to hear your advice. – licheng Oct 08 '21 at 04:16
  • Yes, from the combinatorial data defining the graph, you construct a CW complex, which is a topological space. Take a look, for instance, in Hatcher's book "Algebraic Topology" which is freely available from his home page. – Moishe Kohan Oct 08 '21 at 04:18
  • @Randall there are simple examples of non-isomorphic graphs which are, as topologies, homeomorphic. And time a node has valence $2$ you can remove the node and turn the two edges into one. Basically, there’s nothing distinctive about nodes as points in the topological space. If two graphs have no nodes of valence $2$ then a homeomorphism is a graph isomorphism. This is because the nodes are the points in the topology where no neighborhood is homeomorphic to $(0,1).$ – Thomas Andrews Oct 08 '21 at 04:21
  • @MoisheKohan Thanks for your advice! Actually my question is what is an open set of a graph as a topological space? What is the topology of open sets for the graph. After reading this book, I think the book you recommended should be able to answer my doubts. Thanks again. – licheng Oct 08 '21 at 04:23
  • Definitely not just surfaces, but all topologies of finite graphs can be realized in $\mathbb R^3$ – Thomas Andrews Oct 08 '21 at 04:24
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    That's why I asked you if you know the definition of the quotient topology. The graph $G$ as a topological space, is obtained as the quotient of the disjoint union (coproduct) of $V(G)$ and $E(G)\times [0,1]$, via a certain equivalence relation. Once you understand the definition, you will see what the open subsets are. – Moishe Kohan Oct 08 '21 at 04:41
  • @ThomasAndrews It seems that we first need to clarify the concept of what topological isomorphism of a graph is.
    In graph theory. When you replied to the question of Randall, your meaning seems to be like this: first define the homeomorphism of the two graphs： Two graphs G andG 'are homeomorphic if there is a graph isomorphism from some Subdivision of G to some subdivision of G '. And then define the topological isomorphism of the graphs via on equivalence relation.     – licheng Oct 08 '21 at 04:41
  • @ThomasAndrews As far as I know, the definition of topological isomorphism comes from the textbook Diestel. Of course, it is only defined on the plane graph, which is quite different from what you said：” we call a topological isomorphism $\sigma$ between the plane graphs $G_1$ and $G_2$ if there exists a homeomorphism $\psi : S^2 \rightarrow S^2$ such that $\psi= \pi\varphi\pi^{-1}$ induces $\sigma$ on $V \cup E$. " （that may refer maps one drawing of $G_1$ to the $G_2$）. This definition of topological isomorphism seems to depend on graph drawing . – licheng Oct 08 '21 at 05:02
  • @MoisheKohan Thank you again. I will read the literature you recommended. – licheng Oct 08 '21 at 05:09

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