Editing Disk (mathematics) (section)

==Properties==
The disk has [[circular symmetry]].<ref>{{Cite book|url=https://archive.org/details/iconssymmetries0000altm|url-access=registration|quote=disc circular symmetry.|title=Icons and Symmetries|last=Altmann|first=Simon L.|date=1992|publisher=Oxford University Press|isbn=9780198555995|language=en}}</ref>

The open disk and the closed disk are not topologically equivalent (that is, they are not [[homeomorphism|homeomorphic]]), as they have different topological properties from each other. For instance, every closed disk is [[compact space|compact]] whereas every open disk is not compact.<ref>{{citation|title=New Foundations for Physical Geometry: The Theory of Linear Structures|first=Tim|last=Maudlin|publisher=Oxford University Press|year=2014|isbn=9780191004551|page=339|url=https://books.google.com/books?id=kEbbAgAAQBAJ&pg=PA339}}.</ref> However from the viewpoint of [[algebraic topology]] they share many properties: both of them are [[contractible space|contractible]]<ref>{{citation|title=Combinatorial Group Theory: A Topological Approach|volume=14|series=London Mathematical Society Student Texts|first=Daniel E.|last=Cohen|publisher=Cambridge University Press|year=1989|isbn=9780521349369|url=https://books.google.com/books?id=STc4AAAAIAAJ&pg=PA79|page=79}}.</ref> and so are [[homotopy equivalent]] to a single point. This implies that their [[fundamental group]]s are trivial, and all [[homology group]]s are trivial except the 0th one, which is isomorphic to '''Z'''. The [[Euler characteristic]] of a point (and therefore also that of a closed or open disk) is 1.<ref>In higher dimensions, the Euler characteristic of a closed ball remains equal to +1, but the Euler characteristic of an open ball is +1 for even-dimensional balls and &minus;1 for odd-dimensional balls. See {{citation|title=Introduction to Geometric Probability|first1=Daniel A.|last1=Klain|first2=Gian-Carlo|last2=Rota|author2-link=Gian-Carlo Rota|pages=46–50|publisher=Cambridge University Press|series=Lezioni Lincee|year=1997}}.</ref>

Every [[continuous map]] from the closed disk to itself has at least one [[fixed point (mathematics)|fixed point]] (we don't require the map to be [[bijective]] or even [[surjective]]); this is the case ''n''=2 of the [[Brouwer fixed-point theorem]].<ref>{{harvtxt|Arnold|2013}}, p.&nbsp;132.</ref> The statement is false for the open disk:<ref>{{harvtxt|Arnold|2013}}, Ex.&nbsp;1, p.&nbsp;135.</ref>

Consider for example the function
<math>f(x,y)=\left(\frac{x+\sqrt{1-y^2}}{2},y\right)</math>
which maps every point of the open unit disk to another point on the open unit disk to the right of the given one. But for the closed unit disk it fixes every point on the half circle <math>x^2 + y^2 = 1 , x >0 .</math>