Wikipedia
The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d} with the following non-Hausdorff topology:
{{a, b, c, d}, {a, b, c}, {a, b, d}, {a, b}, {a}, {b}, ∅}
. This topology corresponds to the partial order a < c, b < c, a < d, b < d 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 T. However, from the viewpoint of algebraic topology X has the remarkable property that it is indistinguishable from the circle S.
More precisely the continuous map f from S to X (where we think of S as the unit circle in R) given by
$$f(x,y)=\begin{cases}a\quad x<0\\b\quad x>0\\c\quad(x,y)=(0,1)\\d\quad(x,y)=(0,-1)\end{cases}$$
is a weak homotopy equivalence, that is f induces an isomorphism on all homotopy groups. It follows (proposition 4.21 in Hatcher) 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, 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, 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 which has the same weak homotopy type as the geometric realization |K| of K. More precisely there is a functor, taking K to X, from the category of finite simplicial complexes and simplicial maps and a natural weak homotopy equivalence from |K| to X.