TITLE: In a graph, connectedness in graph sense and in topological sense
QUESTION [6 upvotes]: 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.
As a consequence, a notion of connectedness can be formulated
  independently of the topology on a space. To wit, there is a category
  of connective spaces consisting of sets with collections of connected
  subsets satisfying connectivity axioms; their morphisms are those
  functions which map connected sets to connected sets (Muscat &
  Buhagiar 2006). Topological spaces and graphs are special cases of
  connective spaces; indeed, the finite connective spaces are precisely
  the finite graphs.
However, every graph can be canonically made into a topological space,
  by treating vertices as points and edges as copies of the unit
  interval (see topological graph theory#Graphs as topological spaces).
  Then one can show that the graph is connected (in the graph
  theoretical sense) if and only if it is connected as a topological
  space.

I was wondering if the two bold sentences contradict each other?
Thanks and regards!

REPLY [2 votes]: Actually the 5-cycle (as points) does not admit ANY topology (Hausdorff or not) which gives the same connected sets (ie the edges and paths). This follows by considering that each non-edge induces two open sets. One can prove that each open set will have to be either a set {x} or {y,x,z} where y-x-z are connected, with each type alternating. This implies in turn that any odd cycle cannot have a compatible topology (except 1,3).