The main goal of this paper is to construct universal spaces for asymptotic dimension by generalizing to the coarse context an approach to constructing universal spaces for covering dimension using a factorization result due to Mardesic.

We describe topological obstructions (involving periodic points, topological entropy and rotation sets) for a homeomorphism on a compact manifold to embed in a continuous flow. We prove that homeomorphisms in a $C^{0}$-open and dense set of homeomorphisms isotopic to the identity in compact manifolds of dimension at least two are not the time-1 map of a continuous flow. Such property is also true for volume-preserving homeomorphisms in compact manifolds of dimension at least five. In the case of conservative homeomorphisms of the torus $\mathbb {T}^{d} (d\ge 2)$ isotopic to identity, we describe necessary conditions for a homeomorphism to be flowable in terms of the rotation sets.

Finite subset spaces of a metric space $X$ form a nested sequence under natural isometric embeddings $X=X(1)\subset X(2)\subset \cdots \,$. We prove that this sequence admits Lipschitz retractions $X(n)\rightarrow X(n-1)$ when $X$ is a Hilbert space.

We show that for the Sorgenfrey line S, the minimal dense linearly ordered extension of S is a D-space, but not a monotone D-space; the minimal closed linearly ordered extension of S is not a monotone D-space; the monotone D-property is inversely preserved by finite-to-one closed mappings, but cannot be inversely preserved by perfect mappings.

For every infinite cardinal k we construct a space Ck universal for all compact T1 -spaces of weight ≤ k. It follows, in particular, that there are only 2k topologically different compact T1 -spaces. We show that Cw is universal for all second countable developable T1 -spaces. The existence of closely universal compact T1 -spaces is discussed.

This paper is a continuation of [3] which initiated a systematic study of sufficient conditions for the weighted interpolation inequality of sum form, 1.1 to hold. Here ϕ, θ are non-negative functions of m, j, p, q, r, Ω is a bounded or unbounded domain in Rn, ∊ belongs to an interval Γ=(0, ∊0), u is in a certain Banach space E(Ω), and N, W, P are measurable real functions satisfying N≧ 0, W, P > 0, as well as additional conditions stated below. Finally the constant K does not depend on u although it may depend on the other parameters.

It is shown in this paper that an immersion of a connected, closed n-manifold into another connected n-manifold is a diffeomorphism if and only if the induced homomorphism between the fundamental groups is surjective at some point. This is proved as a consequence of a more general assertion about topological spaces.