Another homogeneous, non-bihomogeneous Peano continuum

Borel Measurable Hahn-Mazurkiewicz Theorem

It is well known due to Hahn and Mazurkiewicz that every Peano continuum is a continuous image of the unit interval. We prove that an assignment, which takes as an input a Peano continuum and produces as an output a continuous mapping whose range is the Peano continuum, can be realized in a Borel measurable way. Similarly, we find a Borel measurable assignment which takes any nonempty compact metric space and assigns a continuous mapping from the Cantor set onto that space. To this end we use the Burgess selection theorem. Finally, a Borel measurable way of assigning an arc joining two selecte…

Hyperspaces of continua with connected boundaries in $π$-Euclidean Peano continua

Let $X$ be a nondegenerate Peano unicoherent continuum. The family $CB(X)$ of proper subcontinua of $X$ with connected boundaries is a $G_δ$-subset of the hyperspace $C(X)$ of all subcontinua of $X$. If every nonempty open subset of $X$ contains an open subset homeomorphic to $\mathbb R^n$ (such space is called $π$-$n$-Euclidean) and $2\le n<\infty$, then $C(X)\setminus CB(X)$ is recognized as an $F_σ$-absorber in $C(X)$; if additionally, no one-dimensional subset separates $X$, then the family of all members of $CB(X)$ which separate $X$ is a $D_2(F_σ)$-absorber in $C(X)$, where $D_2(F_σ)$ de…

A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions

We prove the existence of a 2-dimensional nonaspherical simply connected cell-like Peano continuum (the space itself was constructed in one of our earlier papers). We also indicate some relations between this space and the well-known Griffiths' space from the 1950's.

拓扑分形再探

Topological fractals revisited

本文证明具有不可数多个局部割点的Peano连续统均为拓扑分形，推广了近期结果并部分解答Hata猜想，同时探讨了所需映射的数量。

Alexandroff Manifolds and Homogeneous Continua

We prove the following result announced in Todorov and Valov: Any homogeneous, metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq 1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum with $\check{H}^n(X;G)\neq 0$ is a $V^n$-continuum in the sense of Alexandroff (1957). We also prove that any finite-dimensional homogeneous metric continuum $X$, satisfying $\check{H}^n(X;G)\neq 0$ for some group $G$ and $n\geq 1$, cannot be separated by a compactum $K$ with $\check{H}^{n-1}(K;G)=0$ and …

The topology of systems of hyperspaces determined by dimension functions

Given a non-degenerate Peano continuum $X$, a dimension function $D:2^X_*\to[0,\infty]$ defined on the family $2^X_*$ of compact subsets of $X$, and a subset $Γ\subset[0,\infty)$, we recognize the topological structure of the system $(2^X,\D_{\leγ}(X))_{α\inΓ}$, where $2^X$ is the hyperspace of non-empty compact subsets of $X$ and $D_{\leγ}(X)$ is the subspace of $2^X$, consisting of non-empty compact subsets $K\subset X$ with $D(K)\leγ$.

Covering R-trees

We show that every inner metric space X is the metric quotient of a complete R-tree via a free isometric action, which we call the covering R-tree of X. The quotient mapping is a weak submetry (hence, open) and light. In the case of compact 1-dimensional geodesic space X, the free isometric action is via a subgroup of the fundamental group of X. In particular, the Sierpin'ski gasket and carpet, and the Menger sponge all have the same covering R-tree, which is complete and has at each point valency equal to the continuum. This latter R-tree is of particular interest because it is "universal" in…