# What is the relationship between graph homeomorphism and topology homeomophism？

I came across a term called homeomorphism when I studied graph theory. Recently, I found out that the term homeomorphism has a quite different definition in topology. I want to know whether there is any connection between these two concepts in different areas since the name “homeomorphism” is the same. Can homeomorphism between graphs be understood to be a special form of topology homeomorphism？

## 1 Answer

On any graph you can put a topology by saying that, vaguely speaking, edges should behave like intervals and vertices should be branch points. More concretely, you could define a metric on the graph such that all edges have length one, and use the topology induced by the metric as the topology of the graph. Note that this topology does not detect vertices that only have two edges going in and out. Then, a homeomorphism of graphs (which is an isomorphism between subdivisions) is the same as a homeomorphism of this resulting topological space.

Long story short: The two concepts coincide.

• Note: Canonical topology on graph is weak topology, and weak topology is a metric topology only if each vertex is an end point of only finitely many edges. Ref: Hatcher's Algebraic Topology page 83 and excercise 1.A.1. Feb 16, 2020 at 3:52