Pseudocircle

The pseudocircle is the finite topological space "X" consisting of four distinct points $\{a,b,c,d\}$ with the following non-Hausdorff topology::$left\{\{a,b,c,d\},\{a,b,c\},\{a,b,d\},\{a,b\},\{a\},\{b\},emptyset\; ight\}$"X" is highly pathological from the 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 unit circle $S^1$.

More precisely the map $f:S^1longrightarrow\; X$ 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 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 contractible open sets $\{a\}$ and $\{b\}$.

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 $Kmapsto\; X\_K$ from the category of finite simplicial complexes and simplicial maps and a natural weak homotopy equivalence $|K|longrightarrow\; X\_K$.

References

* "Singular homology groups and homotopy groups of finite topological spaces", by Michael C. McCord, "Duke Math. J.", 33(1966), 465-474.