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Sierpinski carpet

The Sierpinski universal plane curve or the Sierpinski carpet [Sierpinski 1916] M^2_1$ is a well known fractal obtained as the set remaining when one begins with the unit square I^2$ and applies the operation of dividing it into 9 congruent squares and deleting the interior of the central one, then repeats this operation on each of the surviving 8 squares, and so on. See Figure A.

Figure 1.4.2: ( A ) Sierpinski carpet
A.gif

Figure: ( AA ) Sierpinski carpet - an animation
AA.gif

  1. M^2_1$ is universal in the class of all at most one-dimensional subsets of the plane (equivalently, of all boundary subsets of the plane) [Sierpinski 1916], [Sierpinski 1922].
  2. The following statements are equivalent:
    1. X$ is homeomorphic to M^2_1$;
    2. X$ is a locally connected plane curve that contains no local cut points;
    3. X$ is a continuum embeddable in the plane in such a way that \mathbb{R}^2\setminus X$ has infinitely many components C_1, C_2,\dots$ such that diam_i\to 0$, \mathrm{bd}\,C_i\cap \mathrm{bd}\,C_j=\emptyset$ for i\ne j$, \mathrm{bd}\,C_i$ is a simple closed curve for each i$ and \bigcup_{i=1}^\infty \mathrm{bd}\,C_i$ is dense in X$ [Whyburn 1958].
  3. A complete metric space X$ contains a topological copy of M^2_1$ if and only if X$ contains a subset with the bypass property [Prajs 1998a].
  4. The group of all autohomeomorphisms of M^2_1$ has exactly two orbits: one of them is the union of all simple closed curves which are the boundaries of complementary domains of M^2_1$ [Krasinkiewicz 1969].

    The group is a Polish topological group which is totally disconnected and one-dimensional (see [Brechner 1966, Theorem 1.2] and Property 7 in 1.4.1).

  5. Any homeomorphism between Cartesian products of copies of the Sierpinski carpet is factor preserving [Kennedy Phelps 1980]. Consequently, no such product is homogeneous.

  6. The Sierpinski carpet can be continuously decomposed into pseudo-arcs such that the decomposition space is homeomorphic to the carpet [Prajs 1998b, Corollary 18], [Seaquist 1995]. In fact, M^2_1$ is the only planar locally connected curve admitting such a decomposition [Prajs 1998b, Corollary 18].

  7. The Sierpinski carpet is homogeneous with respect to monotone open mappings [Prajs 1998b, Corollary 24], [Seaquist 1999]. Moreover, every continuum which is locally homeomorphic to M^2_1$ (i. e., M^2_1$-manifold) is homogeneous with respect to monotone open mappings [Prajs 1998b, Theorem 23].

  8. M^2_1$ is homogeneous with respect to the class of simple mappings [Charatonik 1984].

  9. If C$ is a curve, then the set of all mappings f:C\to \mathbb{R}^2$ such that f(C)$ is homeomorphic to M^2_1$ is a residual subset of the space (\mathbb{R}^2)^C$ of all mappings of C$ into \mathbb{R}^2$ with the uniform convergence metric.

    If C(X)$ is the hyperspace of all subcontinua of a compact space X$ and C_1(X)$ its subspace of all curves, then the set

    \displaystyle \{\,(C,f)\in C(X)\times (\mathbb{R}2)^X:$   $f(C)$ is homeomorphic to $M^2_1$\displaystyle \,\}$

    is residual in C(X)\times (\mathbb{R}^2)^X$; in other words, almost all mappings in (\mathbb{R}^2)^X$ map almost all curves in X$ onto copies of M^2_1$, where "almost all" means all with except of a subset of the first category in corresponding spaces [Mazurkiewicz 1938].

  10. If a locally compact space X$ contains a topological copy of the Sierpinski carpet, then the space of all copies of the Sierpinski carpets in X$ with the Hausdorff metricHausdorff metric is a true absolute F_{\sigma\delta}$-set [Krupski XXXXb].

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next up previous contents index
Next: Menger universal curve Up: Cyclic examples of locally Previous: Menger universal continua
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30