To say that topology is about stretchy shapes is wrong. Algebraic topology may stretch and twist pieces of Euclidean space within higher dimension Euclidean space to form self intersecting surfaces, but that is not all of topology. The foundation of topology is the topological space. The topological space is a formalism for connectedness and locality. 

I like to use graphs to show the duality of topology. We can generate a topological space for a graph. Depending if we wish to emphasize connectedness or locality, we can set either the edges or vertices as interior points, respectively. The others we set as isolated points. I have not found this example in textbooks.

Graph theory is much like topology, but it emphasizes locality rather than connectedness. These spaces naturally comport to the closed set definition of topological spaces. For applied problems, labeled graphs are used. Mapping the labeling over to topology results in an order on the underlying set. The topological spaces generated from diagraphs are doubly ordered. As labelings have an algebraic structure, the orderings must have them as well. This seems to be a different kind of algebraic topology.