In this paper, a new class of generalized separation axioms (briefly, g-Tg-separation axioms) whose elements are called g-Tg,K, g-Tg,F, g-Tg,H, g-Tg,R, g-Tg,N-axioms is defined in terms sets g-Tg-sets) topological spaces Tg-spaces) and the properties characterizations Tg-space endowed with each such discussed. The study shows that g-Tg,F-axiom implies g-Tg,K-axiom, g-Tg,H-axiom g-Tg,F-axiom, g-...

We explore representing the compact subsets of a given represented space by infinite sequences over Plotkin's T. We show that compact Hausdorff spaces with a proper dyadic subbase admit representations of their compact subsets in such a way that compact sets are essentially underspecified points. We can even ensure that a name of an n-element compact set contains at most n occurrences of ⊥.

A dyadic subbase induces a representation of a second-countable regular space as a subspace of the space of infinite sequences of {0, 1,⊥}, which is known as Plotkin’s . We study four properties of dyadic subbases – full-representing, canonically representing, independent, and minimal – which express efficiency properties of the induced representations.

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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.