Matches in DBpedia 2014 for { <http://dbpedia.org/resource/Pseudocircle> ?p ?o. }
Showing items 1 to 22 of
22
with 100 items per page.
- Pseudocircle abstract "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 byis a weak homotopy equivalence, that is f induces an isomorphism on all homotopy groups. It follows (proposition 4.21 in Hatcher) 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.".
- Pseudocircle wikiPageExternalLink topgpds.html.
- Pseudocircle wikiPageExternalLink ATpage.html.
- Pseudocircle wikiPageID "3441237".
- Pseudocircle wikiPageRevisionID "598015536".
- Pseudocircle hasPhotoCollection Pseudocircle.
- Pseudocircle subject Category:Algebraic_topology.
- Pseudocircle subject Category:Topological_spaces.
- Pseudocircle type Abstraction100002137.
- Pseudocircle type Attribute100024264.
- Pseudocircle type MathematicalSpace108001685.
- Pseudocircle type Set107999699.
- Pseudocircle type Space100028651.
- Pseudocircle type TopologicalSpaces.
- Pseudocircle comment "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.".
- Pseudocircle label "Pseudocircle".
- Pseudocircle sameAs m.09ckz1.
- Pseudocircle sameAs Q7254653.
- Pseudocircle sameAs Q7254653.
- Pseudocircle sameAs Pseudocircle.
- Pseudocircle wasDerivedFrom Pseudocircle?oldid=598015536.
- Pseudocircle isPrimaryTopicOf Pseudocircle.