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The word topology is often used with network, but it is also a branch of mathematics where distance between two connected points doesnt matter as long as their is no holes between them so by that contrast, a donut and coffee cup are same objects in topology.

Question: Is the term "topology" is used in the same context between the two examples?

uhoh
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user0193
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    Welcome to Mathematics Stack Exchange. Yes, in a network, what matters is what is connected to what, not what the path of connection looks like – J. W. Tanner Sep 27 '20 at 19:34
  • @J.W.Tanner Thanks for the greetings! And thanks for the answer as well :) So in geometric topology when we say stretching and compression invariant we mean that the "path" doesnt matter because the path gets stretched and compressed. right? – user0193 Sep 27 '20 at 19:40
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    I'm not an expert in either fields, but I have read a book on network theory and it kind of annoyed me too, that the author just threw the word "topology" around with seemingly no topological space. For me it was incredibly satisfying to find out that simple undirected graphs are topological spaces. This might not be the answer you were looking for, but I am very enthusiastic about this. See on ncatlab for more details https://ncatlab.org/nlab/show/graph#undirected_graphs_as_1complexes_barycentric_subdivision – Máté Kadlicskó Sep 27 '20 at 19:47
  • @MátéKadlicskó it makes so much sense if we say undirected graphs are topological spaces, I know people do throw this work in network without proper introduction. Thanks for the reference! – user0193 Sep 27 '20 at 20:01
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    Of course, the topological space isn't enough by itself to characterize the undirected graph: for example, if $L_n$ is the "line graph" with $n$ vertices and $n-1$ edges (one each between vertex $i$ and $i+1$ for $1 \le i < n$), then the topological spaces for each $L_n$ are all homeomorphic, but the $L_n$ are certainly not isomorphic graphs. – Daniel Schepler Sep 27 '20 at 21:18
  • @DanielSchepler so when we say network in network systems it indeed does pertain to undirected graph, so are you saying that this term network "topology" is a loose term? – user0193 Sep 27 '20 at 22:29
  • The connection is made more precise in https://en.wikipedia.org/wiki/Graph_(topology) and https://en.wikipedia.org/wiki/Topological_graph_theory#Graphs_as_topological_spaces – the gods from engineering Mar 11 '21 at 14:15
  • And I think this was asked here before, in somewhat more precise terms https://math.stackexchange.com/questions/1790823/why-is-the-topology-of-a-graph-called-a-topology – the gods from engineering Mar 11 '21 at 14:22
  • @Fizz thanks it is very much in detail – user0193 Mar 11 '21 at 14:45

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