# topological realizations and graphs

In the following I'm asking not for hand-waving conceptual explanations but for something a bit more formal about graphs and topological realizations.

My understanding is that these objects are functions that takes as input a specific graph and returns a topological space based on that graph, that is called "also" the topological realization of that graph. So, here comes the first question:

• Q0: What is a topological realization of a graph? what is the topology of it?
• Q1: Is there only one definition? or it could be many depends on what?
• Q2: What kind of properties the topological realization of a graph and the graph itself share?
• Q3: How can I prove that if the top. real. of a graph is a finite top. space, then the graph is also finite? If so, how? it seems true, but I don't know where to start with.

Here's the way I would turn a graph into a topological space:

1. Start with a point $$P_v$$ for every vertex $$v$$.
2. Take an interval $$I_{vw}$$ for every edge $$vw$$, but identify one endpoint of $$I_{vw}$$ with $$P_v$$ and the other with $$P_w$$.

I'm sure there are other ways to do it if you try to think of them hard enough, but this seems the most natural.

With this approach, finite graphs do not give finite topological spaces (and I struggle to come up with a good way to represent any graph by a finite topological space). However, finite graphs are precisely the ones that correspond to compact topological spaces.

• What is the topology in this topological space you're proposing? Can we have some other where the topology space is also finite?
– XRow
Commented Nov 23, 2020 at 18:43
• The topology is the quotient topology on the disjoint union of the set of points $P_v$ and the set of intervals $I_{vw}$, using the identifications described to define the quotient. Commented Nov 23, 2020 at 19:33