4 ms·

For people who have time to burn being uselessly pedantic, is this a topologist's world map up to homotopy equivalence or homeomorphism? [edit] I think I get

For people who have time to burn being uselessly pedantic, is this a topologist's world map up to homotopy equivalence or homeomorphism? [edit] I think I get how to encode a hypergraph as a topological space. Let S be a set consisting of two kinds of points: Nodes and hyperedges. Endow it with a topology whose base consists of singleton sets containing only single nodes, and sets containing a hyperedge together with all the nodes it's connected to. In particular, a hyperedge cannot be in an open set unless it's accompanied by the nodes it's connected to. Observations are: The resulting space is usually not T2 or T1, but is always T0; connectivity of the hypergraph is equivalent to connectivity of the topological space; the Sierpinski space results from using a graph with one node and edge. I don't know what that accomplishes except validate the intuition that graph theory is somehow connected to topology.

ouid 4y ago
Homotopy equivalence of what? This isnt really a topological space. It is a pair of spaces, one of which is contained in the other, (the sphere and the borders). This changes the category, morphisms are now pairs of morphisms and must preserve inclusion. The answer is neither, since there is no continuous map that increases the number of connected components of a space and there are more islands than show up on the map. There are other obstacles as well, alaska adds a piece to canada's boundary, this changes the fundamental group of the subspace, which is an invariant of homotopy type, and so the reduced version cannot be homotopy equivalent to the subspace with alaska's boundary included.