This article continues the study of computable elementary topology started in [7]. We introduce a number of computable versions of the topological T0 to T3 separation axioms and solve their logical relation completely. In particular, it turns out that computable T1 is equivalent to computable T2. The strongest axiom SCT3 is used in [2] to construct a computable metric.

Let β G X denote the maximal equivariant compactiίication (G-com-pactίfication) of the G-space X (i.e. a topological space X, completely regular and Hausdorff, on which the topological group G acts as a continuous transformation group). If G is locally compact and locally connected, then we show that β G (XX Y) = β G X X β G Y if and only if X X Y is what we call G-pseudocompact, provided X and...

the knowledge of bygone times of a nation and the discovery of the way through which their thought and culture and civilization developed throughout ages, are of great important for the next generations of that nation and their success in future ways of development. unquestionably, the literary works contain the greatest part of the cultural and mental products and legacy of a nation and deserv...

In this paper, we introduce a new entropy-like invariant, named Hausdorff metric entropy, for finitely generated semigroups acting on compact metric spaces from a set-valued view and study its properties. We establish the relation between Hausdorff metric entropy and topological entropy of a semigroup defined by Bis. Some examples with positive or zero Hausdorff metric entropy are given. Moreov...

A major concept in differentiable dynamics is the Lyapunov exponents of a given map f . It combines the results of ergodic theory with differential properties of f . Consider the following two closely related examples which motivate our paper. Let M be a compact surface and f : M → M a smooth diffeomorphism. Let E denote the set of Borel f -invariant ergodic measures on M . Assume that μ ∈ E . ...

We study the case of an Axiom A holomorphic non-degenerate (hence non-invertible) map f : PC → PC, where PC stands for the complex projective space of dimension 2. Let Λ denote a basic set for f of unstable index 1, and x an arbitrary point of Λ; we denote by δ(x) the Hausdorff dimension of W s r (x)∩Λ, where r is some fixed positive number and W s r (x) is the local stable manifold at x of siz...

we introduce some new concepts of topological spaces which say α − separable topologicalspace and o-topological group, α − first axiom, α − second axiom, and we find some relations betweenthem with some applications in normed spaces.