This is a follow-up to this question: In a graph, connectedness in graph sense and in topological sense
From Wikipedia
Graphs have path connected subsets, namely those subsets for which every pair of points has a path of edges joining them. But it is not always possible to find a topology on the set of points which induces the same connected sets. The 5-cycle graph (and any n-cycle with n>3 odd) is one such example.
I cannot understand this at all. Surely we are talking about putting a topology on the vertex set of the graph. So if we have a path connected graph, the claim seems to be that it is impossible (for some graphs) to find a topology which is also connected. But what's wrong with the indiscrete topology? I would argue that this is a silly topology-- I'd want to topologically distinguish points. But then, if the vertices of the graph are $\{a,b,c,d,\cdots\}$ then try the topology $\tau=\{ \{a,b,c,d,\cdots\}, \{b,c,d,\cdots\}, \{c,d,\cdots\}, \cdots \}$ seems to distinguish points, and be connected.
So maybe the quote implicitly assumes Hausdorff or something? Then how can it possibly only be true for $n$-cycles with "$n>3$ odd"??
I can only assume I've made a major misreading of this quote...?