Pseudocircle Explained

Pseudocircle should not be confused with Quasicircle.

The pseudocircle is the finite topological space X consisting of four distinct points with the following non-Hausdorff topology:$$\backslash .$$

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.f from S¹ to X (where we think of S¹ as the unit circle in \Reals ^{2} ) given byis a weak homotopy equivalence, that is f induces an isomorphism on all homotopy groups. It follows^[1] 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 and whose intersection is also the union of two disjoint contractible open sets and . 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]

Notes and References

1. Allen Hatcher (2002) Algebraic Topology, Proposition 4.21, Cambridge University Press
2. Ronald Brown (2006) "Topology and Groupoids", Bookforce
3. McCord . Michael C. . 1966 . Singular homology groups and homotopy groups of finite topological spaces . . 10.1215/S0012-7094-66-03352-7 . 33 . 465–474.