The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d } with the following non-Hausdorff topology:
This topology corresponds to the partial order
X is highly pathological from the usual viewpoint of general topology, as it fails to satisfy any separation axiom besides T0.
However, from the viewpoint of algebraic topology, X has the remarkable property that it is indistinguishable from the circle S1.
induces an isomorphism on all homotopy groups.
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).
Like S1, X is the union of two contractible open sets {a,b,c} and {a,b,d } whose intersection {a,b} is also the union of two disjoint contractible open sets {a} and {b}.
[2] 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.