Pseudocircle

Article summary

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\}, \varnothing\}. This topology corresponds to the partial order a where the 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 T 0 . However, from the viewpoint of algebraic topology, X has the remarkable property that it is indistinguishable from the circle S 1 . More precisely, the continuous map f from S 1 to X (where we think of S 1 as the unit circle in \Reals^2 ) given by f(x,y) = \begin{cases}a,& x 0\\ c,& (x,y)=(0,1)\\ d,& (x,y)=(0,-1)\end{cases} 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 ordinary or extraordinary homology and cohomology theories (e.g., K-theory). This can be proven 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 disjoint contractible open sets {a} and {b}. So, like S 1 , the result follows from the groupoid Seifert-van Kampen theorem, as in the book Topology and Groupoids.…

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