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[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 {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 XK which has the same weak homotopy type as the geometric realization |K| of K. More precisely there is a functor, taking K to XK, from the category of finite simplicial complexes and simplicial maps and a natural weak homotopy equivalence from |K| to XK.[3]
See also
List of topologies
References
Allen Hatcher (2002) Algebraic Topology, Proposition 4.21, Cambridge University Press
Ronald Brown (2006) "Topology and Groupoids", Bookforce
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
