Pseudocircle

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

${\displaystyle \left\{\{a,b,c,d\},\{a,b,c\},\{a,b,d\},\{a,b\},\{a\},\{b\},\emptyset \right\}}$. 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 f from S¹ to X (where we think of S¹ as the unit circle in R²) given by

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

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 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.

References

1. McCord, Michael C. (1966). "Singular homology groups and homotopy groups of finite topological spaces". Duke Mathematical Journal. 33: 465–474. doi:10.1215/S0012-7094-66-03352-7.
2. Algebraic Topology, by Allen Hatcher, Cambridge University Press, 2002.
3. Ronald Brown, "Topology and Groupoids", Bookforce (2006). Available from amazon.