Pseudocircle

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 formula_2 where open sets are downward closed sets. "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 "S"1.
More precisely the continuous map "f" from "S"1 to "X" (where we think of "S"1 as the unit circle in R2) given by
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 "S"1, "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"}. So like "S"1, 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 "X""K" which has the same weak homotopy type as the geometric realization |"K"| of "K". More precisely there is a functor, taking "K" to "X""K", from the category of finite simplicial complexes and simplicial maps and a natural weak homotopy equivalence from |"K"| to "X""K".