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## Sierpinski carpet

The Sierpinski universal plane curve or the Sierpinski carpet [Sierpinski 1916] is a well known fractal obtained as the set remaining when one begins with the unit square 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.

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. is homeomorphic to ;
2. is a locally connected plane curve that contains no local cut points;
3. is a continuum embeddable in the plane in such a way that has infinitely many components such that , for , is a simple closed curve for each and is dense in [Whyburn 1958].
3. A complete metric space contains a topological copy of if and only if contains a subset with the bypass property [Prajs 1998a].
4. The group of all autohomeomorphisms of has exactly two orbits: one of them is the union of all simple closed curves which are the boundaries of complementary domains of [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, 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 (i. e., -manifold) is homogeneous with respect to monotone open mappings [Prajs 1998b, Theorem 23].

8. is homogeneous with respect to the class of simple mappings [Charatonik 1984].

9. If is a curve, then the set of all mappings such that is homeomorphic to is a residual subset of the space of all mappings of into with the uniform convergence metric.

If is the hyperspace of all subcontinua of a compact space and its subspace of all curves, then the set

$f(C)$ is homeomorphic to $M^2_1$

is residual in ; in other words, almost all mappings in map almost all curves in onto copies of , 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 contains a topological copy of the Sierpinski carpet, then the space of all copies of the Sierpinski carpets in with the Hausdorff metricHausdorff metric is a true absolute -set [Krupski XXXXb].

Here you can find source files of this example.

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Here you can read Notes or write to Notes ies of individual continua.

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