Classification Of Topological Spaces Research Articles

Fixed point logics and definable topological properties

AbstractModal logic enjoys topological semantics that may be traced back to McKinsey and Tarski, and the classification of topological spaces via modal axioms is a lively area of research. In the past two decades, there has been interest in extending topological modal logic to the language of the mu-calculus, but previously no class of topological spaces was known to be mu-calculus definable that was not already modally definable. In this paper, we show that the full mu-calculus is indeed more expressive than standard modal logic, in the sense that there are classes of topological spaces (and weakly transitive Kripke frames), which are mu-definable but not modally definable. The classes we exhibit satisfy a modally definable property outside of their perfect core, and thus we dub them imperfect spaces. We show that the mu-calculus is sound and complete for these classes. Our examples are minimal in the sense that they use a single instance of a greatest fixed point, and we show that least fixed points alone do not suffice to define any class of spaces that is not already modally definable.

The radical-annihilator monoid of a ring

ABSTRACTKuratowski’s closure-complement problem gives rise to a monoid generated by the closure and complement operations. Consideration of this monoid yielded an interesting classification of topological spaces, and subsequent decades saw further exploration using other set operations. This article is an exploration of a natural analogue in ring theory: a monoid produced by “radical” and “annihilator” maps on the set of ideals of a ring. We succeed in characterizing semiprime rings and commutative dual rings by their radical-annihilator monoids, and we determine the monoids for commutative local zero-dimensional (in the sense of Krull dimension) rings.

On some coarse shape invariants

The author and N. Uglešić have recently introduced a new classification of topological spaces, strictly coarser than the shape type classification. The corresponding coarse shape theory is founded on the coarse shape category. In this paper several topological and shape invariants are considered with respect to the coarse shape. It is shown that the coarse shape domination preserves connectedness, shape dimension, movability, n-movability and strong movability. Further, stability is a coarse shape invariant. Moreover, the coarse shape and shape coincide on the class of stable spaces.

Modifications of uniform bases and classification of topological spaces

The notion of a uniform base, which was introduced by P. S. Alexandroff in 1960, turned out to be deeply connected with various properties of topological spaces. Modifications of this notion have led to new metrizability criteria and new directions in classification of topological spaces. A survey of the related results is given.

Quotients of Primes

This note is motivated by our answer, appearing herein as Theorem 4, to the following question posed in a communication from our colleague William Vince Grounds: Given positive real numbers a V. When X is a topological space then the least cardinality of a dense subset of X is one among several interlocking measures of the 'poverty' of the topology on X. For instance, a topological space with a countable dense subset is called separable. These traits are 'topological invariants', and are fundamental to the classification of topological spaces. It is well-known that lR under the usual topology is separable, and indeed that the denumerable set F(N) is dense in the subspace lR'. Theorem 4 states that F(P') is dense in Rll. If S is a finite subset of RN1 then F(S) is finite, and therefore F(S) fails to be dense in X for every infinite subspace X of 1R.

Concerning the classification of topological spaces from the stand point of the theory of retracts

Concerning the classification of topological spaces from the stand point of the theory of retracts