I was reading the following proof of the claim that every graph can be embedded in $\mathbb{R}^3$:
https://sometimesfun.wordpress.com/2015/08/02/embedding-graphs-in-r3/
At the end, there is the following note:
"Although I called this construction an ’embedding’, for lack of a better term, it, the ’embedding’ isn’t a topological embedding of the graph, i.e. the image isn’t homeomorphic to the CW-complex representing the graph."
I don't understand what this means (probably because I know nothing about CW-complexes). I know that an embedding of a graph is basically a drawing, but can someone please explain to me, at least on an intuitive level, what is the difference between an embedding of a graph as is constructed in the above proof, and a topological embedding of a graph?
Thanks in advance.
For example, consider a graph with infinite valence at one vertex, as a subset of $\mathbb{R}^2.$ There must be a sequence of edges with slopes converging to some value. (The same will be true if the graph is a subset of $\mathbb{R}^3$, but it's easier to visualize in the plane.) Thus you have a subgraph which isn't closed, so this is not a topological embedding, if you endow your graph with the CW topology, which should be thought of as the topology coming only from the cells.