Pseudocircle

The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d&hairsp;} with the following non-Hausdorff topology: $$\{\{a,b,c,d\}, \{a,b,c\}, \{a,b,d\}, \{a,b\}, \{a\}, \{b\}, \varnothing\}.$$

This topology corresponds to the partial order $$a0\\ c,& (x,y)=(0,1)\\ d,& (x,y)=(0,-1)\end{cases}$$ is a weak homotopy equivalence, that is $$f$$ induces an isomorphism on all homotopy groups. It follows that $$f$$ also induces an isomorphism on singular homology and cohomology and more generally an isomorphism on all ordinary or extraordinary homology and cohomology theories (e.g., K-theory).

This can be proved using the following observation. Like S1, X is the union of two contractible open sets {a,b,c} and {a,b,d&hairsp;} whose intersection {a,b} is also the union of two disjoint contractible open sets {a} and {b}. So like S1, the result follows from the groupoid Seifert-van Kampen theorem, as in the book Topology and Groupoids.

More generally McCord has shown that for any finite simplicial complex K, there is a finite topological space XK which has the same weak homotopy type as the geometric realization |K| of K. More precisely there is a functor, taking K to XK, from the category of finite simplicial complexes and simplicial maps and a natural weak homotopy equivalence from |K| to XK.