We introduce some new variations of the notions of being Martin-Löf random where the tests are all clopen sets. We explore how these randomness notions relate to classical randomness notions and to degrees of unsolvability.

We prove, for any state in a conformal field theory defined on a set of boundary manifolds with corresponding classical holographic bulk geometry, that for any bipartition of the boundary into two non-clopen sets, the density matrix cannot be a tensor product of the reduced density matrices on each region of the bipartition. In particular, there must be entanglement across the bipartition surfa...

α is a regular cardinal number or the symbol ∞, and X is a compact Hausdorff space. It is shown that (Theorem 2.1) X is αcozerocomplemented iff each αBorel set differs from an αcozero set by a meagre set. This immediately yields (Corollary 2.2) X is αdisconnected iff each αBorel set differs from a clopen set by a meagre set. (The cases α = ∞ in Theorem 2.1, and α = ω1 in Corollary 2.2 are impor...

Topological fixpoint logics are a family of logics that admits topological models and where the fixpoint operators are defined with respect to the topological interpretations. Here we consider a topological fixpoint logic for relational structures based on Stone spaces, where the fixpoint operators are interpreted via clopen sets. We develop a game-theoretic semantics for this logic. First we i...

‎three infinite families of finite abelian groups will be‎ ‎described such that each member of these families has‎ ‎the r'edei $k$-property for many non-trivial values of $k$‎.

Assume that X is a metrizable separable space, and each clopen-valued lower semicontinuous multivalued map Φ : X ⇒ Q has a continuous selection φ : X → Q. Our main result is that in this case, X is a σ-space. We also derive a partial converse implication, and present a reformulation of the Scheepers Conjecture in the language of continuous selections.

The concept of an attractor is essential for understanding the dynamics of cellular automata. An attractor is defined as a limit set (intersection of forward images) of a nonempty clopen (closed and open) invariant set. Classification of cellular automata based on the system of their attractors has been considered in Hurley [1] or Kůrka [2]. The basic distinction is between cellular automata wh...

Using a system of axioms among with a modified definition of boundary on the basis of the intuitionistic fuzzy sets, we formulate an inductive structure for the dimension of fuzzy spaces which has been defined by Coker. This new definition of boundary allows to characterize an intuitionistic fuzzy clopen set as a set with zero boundary. Also, some critical properties and applications are establ...