For each natural number n ≥ 1 and each pair of ordinals α,β with n ≤ α ≤ β ≤ ω(⁺), where ω(⁺) is the first ordinal of cardinality ⁺, we construct a continuum $S_{n,\alpha ,\beta }$ such that (a) $dim{S}_{n,\alpha ,\beta }=n$; (b) $trDg{S}_{n,\alpha ,\beta }=trDgo{S}_{n,\alpha ,\beta }=\alpha$; (c) $trind{S}_{n,\alpha ,\beta }=trInd₀S_{n,\alpha ,\beta }=\beta$; (d) if β < ω(⁺), then $S_{n,\alpha ,\beta }$ is separable and first countable; (e) if n = 1, then $S_{n,\alpha ,\beta }$ can be made chainable or hereditarily decomposable; (f) if α = β < ω(⁺), then $S_{n,\alpha ,\beta }$ can be made hereditarily indecomposable; (g) if n = 1 and α = β < ω(⁺), then $S_{n,\alpha ,\beta }$ can be made chainable and hereditarily indecomposable. In particular,...

Let $K$ be a closed convex subset of a complete convex metric space $X$ and $T,I:K\to K$ two compatible mappings satisfying following contraction definition: $Tx,Ty)\le (Ix,Iy)+(1-a)max\{Ix.Tx),Iy,Ty)\}$ for all $x,y$ in $K$, where $0<a<1/2^{p-1}$ and $p\ge 1$. If $I$ is continuous and $I\left(K\right)$ contains $\left[T\right(K\left)\right]$ , then $T$ and $I$ have a unique common fixed point in $K$ and at this point $T$ is continuous. This result gives affirmative answers to open questions set forth by Diviccaro, Fisher and Sessa in connection with necessarity of hypotheses of linearity and non-expansivity of $I$ in their Theorem [3]...

We prove that, for every finite-dimensional metrizable space, there exists a compactification that is Eberlein compact and preserves both the covering dimension and weight.

A.V. Arkhangel’skii asked that, is it true that every space $Y$ of countable tightness is homeomorphic to a subspace (to a closed subspace) of $C_p\left(X\right)$ where $X$ is Lindelöf? $C_p\left(X\right)$ denotes the space of all continuous real-valued functions on a space $X$ with the topology of pointwise convergence. In this note we show that the two arrows space is a counterexample for the problem by showing that every separable compact linearly ordered topological space is second countable if it is homeomorphic to a subspace of $C_p\left(X\right)$...

Defining the complexity of a green pattern exhibited by an interval map, we give the best bounds of the topological entropy of a pattern with a given complexity. Moreover, we show that the topological entropy attains its strict minimum on the set of patterns with fixed eccentricity m/n at a unimodal X-minimal case. Using a different method, the last result was independently proved in[11].

We consider a class of symbolic systems over a finite alphabet which are minimal almost one-to-one extensions of rotations of compact metric monothetic groups and provide computations of their enveloping semigroups that highlight their algebraic structure. We describe the set of idempotents of these semigroups and introduce a classification that can help distinguish between certain such systems having zero topological entropy.