A characterisation of trivial 1-cohomology, in terms of some connectedness condition, is presented for a broad class of metric spaces.

Let $C_F\left(X\right)$ be the socle of C(X). It is shown that each prime ideal in $C\left(X\right)/C_F\left(X\right)$ is essential. For each h ∈ C(X), we prove that every prime ideal (resp. z-ideal) of C(X)/(h) is essential if and only if the set Z(h) of zeros of h contains no isolated points (resp. int Z(h) = ∅). It is proved that $dim(C\left(X\right)/C_F\left(X\right))\ge dimC\left(X\right)$, where dim C(X) denotes the Goldie dimension of C(X), and the inequality may be strict. We also give an algebraic characterization of compact spaces with at most a countable number of nonisolated points. For each essential...

We give a description of possible sets of cycle lengths for distance-decreasing maps and isometries of the ring of n-adic integers.

Michael Handel proved the existence of a fixed point for an orientation preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. More recently, the author generalized Handel's theorem to a wider class of cycles of links. In this paper we complete this topic describing exactly which are all the cycles of links forcing the existence of a fixed point.

In this paper we introduce generalized cyclic contractions through $r$ number of subsets of a probabilistic 2-metric space and establish two fixed point results for such contractions. In our first theorem we use the Hadzic type $t$-norm. In another theorem we use a control function with minimum $t$-norm. Our results generalizes some existing fixed point theorem in 2-Menger spaces. The results are supported with some examples.

The main results of this paper are: 1. No topologically transitive cocycle ${ℝ}^m$-extension of minimal rotation on the unit circle by a continuous real-valued bounded variation ℤ-cocycle admits minimal subsets. 2. A minimal rotation on a compact metric monothetic group does not admit a topologically transitive real-valued cocycle if and only if the group is finite.

It is proved that the cylinder X × I over a λ-dendroid X has the fixed point property. The proof uses results of [9] and [10].