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Menger universal curve

The Menger universal curve M^3_1$ is the subset of the unit cube I^3$ whose projections onto faces of the cube are the Sierpinski carpets, i.e.,

\displaystyle M^3_1=\{\,(x,y,z)\in I^3: (x,y)\in M^2_1,\quad (y,z)\in M^2_1,
\quad (x,z)\in M^2_1\,\}$

[Menger 1926]. Other descriptions can be found in 1.4.1 (see also [Mayer et al. 1986, pp. 5-6, 8-9]). See Figure A.

Figure 1.4.3: ( A ) Menger universal curve

Figure: ( AA ) Menger universal curve - an animation

  1. The following statements are equivalent:
    1. X$ is homeomorphic to M^3_1$;
    2. X$ is a locally connected curve with no local cut points and no planar open nonempty subsets [Anderson 1958, Theorem XII, p. 13];
    3. X$ is a homogeneous locally connected curve different from a simple close curve [Anderson 1958, Theorem XIII, p. 14];
    4. X$ is a locally connected curve with the disjoint arcs property (see Property 3 in 1.4.1);
    5. X$ is a locally connected curve and each arc in X$ is approximately non-locally-separating arc and has empty interior in X$ [Krupski et al. XXXXa, Theorem 3, p. 86].

  2. M^3_1$ is universal in the class of all metric separable spaces of dimension \le 1$ [Menger 1926].

  3. Z-setsZ-set in M^3_1$ coincide with non-locally-separating closed subsets of M^3_1$.

    If Z$ is a separable metric space of dimension \le 1$, then Z$ can be embedded as a non-locally-separating subset of M^3_1$ [Mayer et al. 1986, Theorem 6.1, p. 42].

  4. If Z$ is a closed subset of a metric space X$, \dim(X\setminus Z)\le 1$ and f:Z\to M^3_1$ is a continuous mapping with non-locally separating image f(Z)$, then f$ can be extended to a map g:X\to M^3_1$ such that g\vert X\setminus Z$ is an embedding into M^3_1\setminus f(Z)$ [Mayer et al. 1986, Theorem 6.4, p. 44].

  5. Every continuous surjection h$ between non-locally separating closed subsets Z,Z'$ of M^3_1$ extends to a mapping h^*$ of M^3_1$ such that h^*\vert M^3_1\setminus Z$ is a homeomorphism onto M^3_1\setminus Z'$ [Mayer et al. 1986, Corollary 6.5, p. 44]; moreover, for every \epsilon>0$ there exists \delta>0$ such that if h$ is a \delta $-homeomorphism, then h^*$ can be taken as an \epsilon$-homeomorphism (see Property 4b in 1.4.1).

  6. For each locally connected continuum X$, there exist open surjections f:M^3_1\to X$ and g:M^3_1\to X$ such that f^{-1}(x)$ is homeomorphic to M^3_1$ and g^{-1}(x)$ is a Cantor set, for any x\in X$ [Wilson 1972].

  7. Any compact 0-dimensional group G$ acts freely on M^3_1$ so that the orbit space M^3_1/G$ is homeomorphic to M^3_1$ [Anderson 1957, Theorem 1].

  8. If a locally compact space X$ contains a topological copy of M^3_1$, then the space of all copies of M^3_1$ in X$ with the Hausdorff metricHausdorff metric is a true absolute F_{\sigma\delta}$-set [Krupski XXXXb].

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Next: 2-dimensional locally connected continua Up: Cyclic examples of locally Previous: Sierpinski carpet
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih