Editing Smith–Volterra–Cantor set

{{Short description|Set of real numbers in mathematics}}
[[Image:Smith-Volterra set.png|thumb|right|256px|After black intervals have been removed, the white points which remain are a nowhere dense set of measure 1/2.]] 
In [[mathematics]], the '''Smith–Volterra–Cantor set''' ('''SVC'''), '''ε-Cantor set''',<ref>Aliprantis and Burkinshaw (1981), Principles of Real Analysis</ref> or '''fat Cantor set''' is an example of a set of points on the [[real line]] that is [[nowhere dense]] (in particular it contains no [[interval (mathematics)|interval]]s), yet has positive [[measure (mathematics)|measure]]. The Smith–Volterra–Cantor set is named after the [[mathematician]]s [[Henry John Stephen Smith|Henry Smith]], [[Vito Volterra]] and [[Georg Cantor]].  In an 1875 paper, Smith discussed a nowhere-dense set of positive measure on the real line,<ref>[[Henry John Stephen Smith|Smith, Henry J.S.]] (1874). "[https://babel.hathitrust.org/cgi/pt?id=ucm.5324906759;view=1up;seq=148 On the integration of discontinuous functions]". Proceedings of the London Mathematical Society. First series. 6: 140–153
</ref> and Volterra introduced a similar example in 1881.<ref>{{Cite journal |last1=Ponce Campuzano |first1=Juan |last2=Maldonado |first2=Miguel |date=2015 |title=Vito Volterra's construction of a nonconstant function with a bounded, non Riemann integrable derivative |journal=BSHM Bulletin Journal of the British Society for the History of Mathematics |volume=30 |issue=2 |pages=143–152 |doi=10.1080/17498430.2015.1010771|s2cid=34546093 }}</ref>  The Cantor set as we know it today followed in 1883. The Smith–Volterra–Cantor set is [[Homeomorphism|topologically equivalent]] to the [[Cantor set|middle-thirds Cantor set]].

==Construction==

Similar to the construction of the [[Cantor set]], the Smith–Volterra–Cantor set is constructed by removing certain intervals from the [[unit interval]] <math>[0, 1].</math> 

The process begins by removing the middle 1/4 from the interval <math>[0, 1]</math> (the same as removing 1/8 on either side of the middle point at 1/2) so the remaining set is 
<math display=block>\left[0, \tfrac{3}{8}\right] \cup \left[\tfrac{5}{8}, 1\right].</math>

The following steps consist of removing subintervals of width <math>1/4^n</math> from the middle of each of the <math>2^{n-1}</math> remaining intervals.  So for the second step the intervals <math>(5/32, 7/32)</math> and <math>(25/32, 27/32)</math> are removed, leaving 
<math display=block>\left[0, \tfrac{5}{32}\right] \cup \left[\tfrac{7}{32}, \tfrac{3}{8}\right] \cup \left[\tfrac{5}{8}, \tfrac{25}{32}\right] \cup \left[\tfrac{27}{32}, 1\right].</math>

Continuing indefinitely with this removal, the Smith–Volterra–Cantor set is then the set of points that are never removed. The image below shows the initial set and five iterations of this process. 

[[Image:Smith-Volterra-Cantor set.svg|center|512px]]

Each subsequent iterate in the Smith–Volterra–Cantor set's construction removes proportionally less from the remaining intervals.  This stands in contrast to the [[Cantor set]], where the proportion removed from each interval remains constant.  Thus, the Smith–Volterra–Cantor set has positive measure while the Cantor set has zero measure.

==Properties==

By construction, the Smith–Volterra–Cantor set contains no intervals and therefore has empty interior. It is also the intersection of a sequence of [[closed set]]s, which means that it is closed.
During the process, intervals of total length 
<math display=block>\sum_{n=0}^\infty \frac{2^n}{2^{2n + 2}} = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots = \frac{1}{2}\,</math> 
are removed from <math>[0, 1],</math> showing that the set of the remaining points has a positive measure of 1/2. This makes the Smith–Volterra–Cantor set an example of a closed set whose [[Boundary (topology)|boundary]] has positive [[Lebesgue measure]].

== Other fat Cantor sets ==

In general, one can remove <math>r_n</math> from each remaining subinterval at the <math>n</math>th step of the algorithm, and end up with a Cantor-like set.  The resulting set will have positive measure if and only if the sum of the sequence is less than the measure of the initial interval. For instance, suppose the middle intervals of length <math>a^n</math> are removed from <math>[0, 1]</math> for each <math>n</math>th iteration, for some <math>0 \leq a \leq \dfrac{1}{3}.</math> Then, the resulting set has Lebesgue measure 
<math display=block>\begin{align}
1 - \sum _{n=0}^\infty 2^n a ^ {n+1} &= 1 - a \sum _{n=0}^\infty (2a)^n \\[5pt]
                                     &= 1 - a \frac{1}{1 - 2a} \\[5pt]
                                     &= \frac{1 - 3a}{1 - 2a}
\end{align}
</math>
which goes from <math>0</math> to <math>1</math> as <math>a</math> goes from <math>1/3</math> to <math>0.</math>  (<math>a > 1/3</math> is impossible in this construction.)

Cartesian products of Smith–Volterra–Cantor sets can be used to find [[totally disconnected space|totally disconnected set]]s in higher dimensions with nonzero measure. By applying the [[Denjoy–Riesz theorem]] to a two-dimensional set of this type, it is possible to find an [[Osgood curve]], a [[Jordan curve]] such that the points on the curve have positive area.<ref>{{citation
 | last1 = Balcerzak | first1 = M.
 | last2 = Kharazishvili | first2 = A.
 | doi = 10.1023/A:1022102312024
 | issue = 3
 | journal = Georgian Mathematical Journal
 | mr = 1679442
 | pages = 201–212
 | title = On uncountable unions and intersections of measurable sets
 | volume = 6
 | year = 1999| doi-broken-date = 17 March 2025
 }}.</ref>

==See also==

* The Smith–Volterra–Cantor set is used in the construction of [[Volterra's function]] (see external link).
* The Smith–Volterra–Cantor set is an example of a [[compact set]] that is not Jordan measurable, see [[Jordan measure#Extension to more complicated sets]].
* The [[indicator function]] of the Smith–Volterra–Cantor set is an example of a bounded function that is not Riemann integrable on (0,1) and moreover, is not equal almost everywhere to a Riemann integrable function, see [[Riemann integral#Examples]].
* {{annotated link|List of topologies}}

==References==

{{reflist|30em}}

{{DEFAULTSORT:Smith-Volterra-Cantor set}}
[[Category:Fractals]]
[[Category:Measure theory]]
[[Category:Sets of real numbers]]
[[Category:Topological spaces]]