Editing Sierpiński carpet (section)

==Properties==
[[File:Peano Sierpinski carpet 4.svg|thumb|Variant of the [[Peano curve]] with the middle line erased creates a Sierpiński carpet]]

The area of the carpet is zero (in standard [[Lebesgue measure]]).
:'''Proof:''' Denote as {{mvar|a<sub>i</sub>}} the area of iteration {{mvar|i}}. Then {{math|''a''<sub>''i'' + 1</sub> {{=}} {{sfrac|8|9}}''a<sub>i</sub>''}}. So {{math|''a<sub>i</sub>'' {{=}} ({{sfrac|8|9}}){{sup|''i''}}}}, which tends to 0 as {{mvar|i}} goes to infinity.

The [[interior (topology)|interior]] of the carpet is empty.
:'''Proof:''' Suppose by contradiction that there is a point {{mvar|P}} in the interior of the carpet. Then there is a square centered at {{mvar|P}} which is entirely contained in the carpet. This square contains a smaller square whose coordinates are multiples of {{math|{{sfrac|1|3<sup>''k''</sup>}}}} for some {{mvar|k}}. But, if this square has not been previously removed, it must have been holed in iteration {{math|''k'' + 1}}, so it cannot be contained in the carpet – a contradiction.

The [[Hausdorff dimension]] of the carpet is <math>\frac{\log 8}{\log 3} \approx 1.8928</math>.<ref>{{cite book | author-link=Stephen Semmes | title=Some Novel Types of Fractal Geometry | series=Oxford Mathematical Monographs | publisher=Oxford University Press | year=2001 | isbn=0-19-850806-9 | zbl=0970.28001 | last=Semmes | first=Stephen | page=31 }}</ref>

Sierpiński demonstrated that his carpet is a universal plane curve.<ref name=sierpinski>{{cite journal | jfm=46.0295.02 | last=Sierpiński | first=Wacław | author-link=Wacław Sierpiński | title=Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée | language=fr | journal=C. R. Acad. Sci. Paris | volume=162 | pages=629–632 | year=1916 | issn=0001-4036 }}</ref> That is: the Sierpiński carpet is a compact subset of the plane with [[Lebesgue covering dimension]] 1, and every subset of the plane with these properties is [[homeomorphic]] to some subset of the Sierpiński carpet.

This "universality" of the Sierpiński carpet is not a true universal property in the sense of category theory: it does not uniquely characterize this space up to homeomorphism.  For example, the disjoint union of a Sierpiński carpet and a circle is also a universal plane curve.  However, in 1958 [[Gordon Whyburn]]<ref name=whyburn>{{cite journal | last=Whyburn | first=Gordon | author-link=Gordon Whyburn | title=Topological chcracterization of the Sierpinski curve | journal=Fund. Math. | volume=45 | pages=320–324 | year=1958 | doi=10.4064/fm-45-1-320-324 | doi-access=free }}</ref> uniquely characterized the Sierpiński carpet as follows: any curve that is [[locally connected]] and has no 'local cut-points' is homeomorphic to the Sierpiński carpet.  Here a '''local cut-point''' is a point {{mvar|p}} for which some connected neighborhood {{mvar|U}} of {{mvar|p}} has the property that {{math|''U'' − {''p''} }} is not connected. So, for example, any point of the circle is a local cut point.

In the same paper Whyburn gave another characterization of the Sierpiński carpet. Recall that a [[Continuum (topology)|continuum]] is a nonempty connected compact metric space. Suppose {{mvar|X}} is a continuum embedded in the plane. Suppose its complement in the plane has countably many connected components {{math|''C''<sub>1</sub>, ''C''<sub>2</sub>, ''C''<sub>3</sub>, ...}} and suppose:

* the diameter of {{mvar|C<sub>i</sub>}} goes to zero as {{math|''i'' → ∞}};
* the boundary of {{mvar|C<sub>i</sub>}} and the boundary of {{mvar|C<sub>j</sub>}} are disjoint if {{math|''i'' ≠ ''j''}};
* the boundary of {{mvar|C<sub>i</sub>}} is a simple closed curve for each {{mvar|i}};
* the union of the boundaries of the sets {{mvar|C<sub>i</sub>}} is dense in {{mvar|X}}.

Then {{mvar|X}} is homeomorphic to the Sierpiński carpet.