The local structure of homogeneous continua (curves) is studied. Components of open subsets of each homogeneous curve which is not a solenoid have the disjoint arcs property. If the curve is aposyndetic, then the components are nonplanar. A new characterization of solenoids is formulated: a continuum is a solenoid if and only if it is homogeneous, contains no terminal nontrivial subcontinua and small subcontinua are not ∞-ods.

A metric space (X,ϱ) satisfies the disjoint (0,n)-cells property provided for each point x ∈ X, any map f of the n-cell ${B}^{n}$ into X and for each ε > 0 there exist a point y ∈ X and a map $g:{B}^{n}\to X$ such that ϱ(x,y) < ε, $\widehat{\varrho}(f,g)<\epsilon $ and $y\notin g\left({B}^{n}\right)$. It is proved that each homogeneous locally compact ANR of dimension >2 has the disjoint (0,2)-cells property. If dimX = n > 0, X has the disjoint (0,n-1)-cells property and X is a locally compact $L{C}^{n-1}$-space then local homologies satisfy ${H}_{k}(X,X-x)=0$ for k < n and Hn(X,X-x) ≠ 0.

It is shown that the following hyperspaces, endowed with the Hausdorff metric, are true absolute ${F}_{\sigma \delta}$-sets:
(1) ℳ ²₁(X) of Sierpiński universal curves in a locally compact metric space X, provided ℳ ²₁(X) ≠ ∅ ;
(2) ℳ ³₁(X) of Menger universal curves in a locally compact metric space X, provided ℳ ³₁(X) ≠ ∅ ;
(3) 2-cells in the plane.

It is shown that for every numbers ${m}_{1},{m}_{2}\in \{3,\cdots ,\omega \}$ there is a strongly self-homeomorphic dendrite which is not pointwise self-homeomorphic. The set of all points at which the dendrite is pointwise self-homeomorphic is characterized. A general method of constructing a large family of dendrites with the same property is presented.

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