https://en.formulasearchengine.com/api.php?action=feedcontributions&feedformat=atom&user=86.29.234.250formulasearchengine - User contributions [en]2026-05-03T22:58:12ZUser contributionsMediaWiki 1.43.0-wmf.28https://en.formulasearchengine.com/index.php?title=Mean_absolute_percentage_error&diff=12162Mean absolute percentage error2013-12-28T23:26:20Z<p>86.29.234.250: Correct MAPE formula</p>
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<div>The '''pseudocircle''' is the [[finite topological space]] ''X'' consisting of four distinct points {''a'',''b'',''c'',''d''} with the following [[Hausdorff space|non-Hausdorff]] topology:<br />
:<math>\left\{\{a,b,c,d\},\{a,b,c\},\{a,b,d\},\{a,b\},\{a\},\{b\},\emptyset\right\}</math>. This topology corresponds to the partial order <math>a<c,b<c,a<d,b<d</math> where open sets are downward closed sets. <br />
''X'' is highly [[pathological (mathematics)|pathological]] from the usual viewpoint of [[general topology]] as it fails to satisfy any [[separation axiom]] besides [[Kolmogorov space|T<sub>0</sub>]]. However from the viewpoint of [[algebraic topology]] ''X'' has the remarkable property that it is indistinguishable from the [[circle]] ''S''<sup>1</sup>.<br />
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More precisely the [[continuous map]] ''f'' from ''S''<sup>1</sup> to ''X'' (where we think of ''S''<sup>1</sup> as the [[unit circle]] in '''R'''<sup>2</sup>) given by<br />
:<math>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}</math><br />
is a [[Whitehead theorem|weak homotopy equivalence]], that is ''f'' induces an isomorphism on all [[homotopy group]]s. It follows (proposition 4.21 in Hatcher) that ''f'' also induces an isomorphism on [[singular homology|singular homology and cohomology]] and more generally an isomorphism on all ordinary or extraordinary [[homology theory|homology and cohomology theories]] (e.g., [[K-theory]]).<br />
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This can be proved using the following observation. Like ''S''<sup>1</sup>, ''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 sets|disjoint]] contractible open sets {''a''} and {''b''}.<br />
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More generally McCord has shown that for any finite [[abstract simplicial complex|simplicial complex]] ''K'', there is a [[finite topological space]] ''X''<sub>''K''</sub> which has the same weak homotopy type as the [[abstract simplicial complex|geometric realization]] |''K''| of ''K''. More precisely there is a [[functor]], taking ''K'' to ''X''<sub>''K''</sub>, from the category of finite simplicial complexes and simplicial maps and a [[natural transformation|natural]] weak homotopy equivalence from |''K''| to ''X''<sub>''K''</sub>.<br />
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== References ==<br />
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# {{cite journal | author=Michael C. McCord | title=Singular homology groups and homotopy groups of finite topological spaces | journal=[[Duke Mathematical Journal]] | year=1966 | volume=33 | pages=465–474 | doi=10.1215/S0012-7094-66-03352-7}}<br />
# ''[http://www.math.cornell.edu/~hatcher/AT/ATpage.html Algebraic Topology]'', by Allen Hatcher, ''Cambridge University Press'', 2002.<br />
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[[Category:Algebraic topology]]<br />
[[Category:Topological spaces]]</div>86.29.234.250