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We present a detailed discussion of the problem of embeddability of connected closed surfaces into products of two curves. In particular, we comment on two important results by R. Cauty and W. Kuperberg in this direction and strengthen them. It is proved that a closed connected surface M lying in a product of two curves is a retract of the product if and only if M is a torus. It is also observed the...
We prove that any infinite-dimensional hereditarily indecomposable compactum can be represented as the limit of an inverse sequence of compacta X0 <-- X1 <-- X2 <--..., where each Xk, k > O, is hereditarily indecomposable, dim Xk = k, and the bonding maps pk k+1 : X k+1 --> Xk are 1-dimensional monotone and surjective. Moreover, each bonding map pk k+l : Xk,l > 1, is l-dimensional...
In this note we discuss symmetric continua in the 2-sphere [S^2], i.e. continua invariant under the antipodal map. Our results indicate an essential difference between decomposable and indecomposable elements in this class. Symmetric decomposable continua can ahvays be written in a natural form [union of sets C and (-C)], where C is a non-symmetric subcontinuum of [S^2] connecting two antipodal points...
First we show that for any mapping f : [S^2] --> R there exist two antipodal points p, -p [belongs to S^2] and a continuum C [is a subset of S^2] connecting them such that f is constant on C (Corollary 3.2)^(1); if f is equivariant (with respect to the canonical involutions) then C can be chosen symmetric (Corollary 3.2 or Lemma 5.4). This, combined with a result about the equalization of mappings...
We prove that for every family of sujective mappings f[sub j] : X[sub j] --> I, j [belongs to] J, where X[sub j] are continua, there exist a continuum Y and surjective mappings g[sub j] : Y --> X[sub j] such that f[sub j] o g[sub j] = f[sub j'] o g[sub j'] for all j, j' [belongs to] J.
We prove a stronger version of the classical theorem of Whyburn on the liftings of dendrites relative to 0-dimensional open mappings of compact metric spaces.
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