Math 2.0

General topology Continua and dimension theory Example

Sierpiński carpet

Builds on Compact spaceConnected spaceLebesgue covering dimension

Example

Let S0=[0,1]2S_0 = [0, 1]^2. Divide each square of side 3−k3^{-k} in SkS_k into 99 equal squares of side 3−k−13^{-k-1}. Remove the interior of the middle one; the result is Sk+1S_{k+1}. The Sierpiński carpet is S=⋂k≥0SkS = \bigcap_{k \ge 0} S_k.

The set SS is a continuum. Each SkS_k is a finite union of closed squares, so it is compact. It is connected, since the 88 kept squares in each divided square form a ring in which neighbours share an edge. The intersection of a decreasing sequence of nonempty compact connected sets in the plane is nonempty, compact and connected.

Remarks

The set SkS_k consists of 8k8^k squares of area 9−k9^{-k}, so SS has area lim⁡k(8/9)k=0\lim_k (8/9)^k = 0. Its Hausdorff dimension is log⁡8/log⁡3≈1.893\log 8 / \log 3 \approx 1.893. Its Lebesgue covering dimension is 11. Sierpiński showed that every compact subset of the plane of dimension at most 11 embeds in the carpet, so it is a universal plane curve.

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Compact space, concept, no evidence levelCompact spaceconceptConnected space, concept, no evidence levelConnected spaceconceptLebesgue covering dimension, concept, no evidence levelLebesgue covering dim…conceptSierpiński carpet, example, no evidence levelSierpiński carpetexample
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  1. Compact space concept
  2. Connected space concept
  3. Lebesgue covering dimension concept
  4. Sierpiński carpet example

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Sierpi%C5%84ski_carpet
About this page
TagPCM3H6J3EE
URIm2://PCM3H6J3EE
Kindexample
WritingW0 stub
MSC54F15, 28A80, 54F50
Instance ofContinuum
UsesCompact space, Connected space, Lebesgue covering dimension
Same as
Changed2 November 2026

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