Pseudocircle

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 T₀. However, from the viewpoint of algebraic topology X has the remarkable property that it is indistinguishable from the circle S¹.

More precisely the continuous map f from S¹ to X (where we think of S¹ as the unit circle in R²) given by

f ( x , y ) = { a x < 0 b x > 0 c ( x , y ) = ( 0 , 1 ) d ( x , y ) = ( 0 , − 1 )

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 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}. 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 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.