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- The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d } with the following non-Hausdorff topology: . This topology corresponds to the partial order 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 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 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 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. (en)
- Псевдоко́ло — скінченний топологічний простір, невідмінний від кола з погляду алгебричної топології. (uk)
- Псевдоокружность — конечное топологическое пространство, неотличимое от окружности с точки зрения алгебраической топологии. (ru)
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- Псевдоко́ло — скінченний топологічний простір, невідмінний від кола з погляду алгебричної топології. (uk)
- Псевдоокружность — конечное топологическое пространство, неотличимое от окружности с точки зрения алгебраической топологии. (ru)
- The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d } with the following non-Hausdorff topology: . This topology corresponds to the partial order 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 (en)
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