Pseudocircle

The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d } with the following non-Hausdorff topology:

$${\displaystyle \{\{a,b,c,d\},\{a,b,c\},\{a,b,d\},\{a,b\},\{a\},\{b\},\varnothing \}.}$$

This topology corresponds to the partial order ${\displaystyle a<c,\ b<c,\ a<d,\ b<d}$ 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 ${\displaystyle f}$ from S¹ to X (where we think of S¹ as the unit circle in ${\displaystyle \mathbb {R} ^{2}}$) given by

$${\displaystyle f(x,y)={\begin{cases}a,&x<0\\b,&x>0\\c,&(x,y)=(0,1)\\d,&(x,y)=(0,-1)\end{cases}}}$$

is a weak homotopy equivalence, that is ${\displaystyle f}$ induces an isomorphism on all homotopy groups. It follows[1] that ${\displaystyle f}$ 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.[2]

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.[3]