Extension Theory: The interface between set-theoretic and algebraic topology

Order-theoretic Invariants in Set-theoretic Topology

We present several results related to van Douwen’s Problem, which asks whether there is homogeneous compactum with cellularity exceeding c, the cardinality of the reals. For example, just as all known homogeneous compacta have cellularity at most c, they satisfy similar upper bounds in terms of Peregudov’s Noetherian type and related cardinal functions defined by order-theoretic base properties...

Part I Set Theoretic Topology

Combining Algebraic and Set-Theoretic Specifications

ion and application. The underlying logic for reasoning about such a speci cation is a powerful higher-order logic, e.g., based on ZF set theory. Property-oriented speci cations. Here one generally tries to avoid choosing an explicit representation: types are left abstract as so-called sorts| sometimes equipped with a subsort inclusion relation, but otherwise unstructured. The operations are sp...

Combining Algebraic and Set-theoretic Speciications ?

Speciication frameworks such as B and Z provide power sets and cartesian products as built-in type constructors, and employ a rich notation for deening (among other things) abstract data types using for-mulae of predicate logic and lambda-notation. In contrast, the so-called algebraic speciication frameworks often limit the type structure to sort constants and rst-order functionalities, and res...

Categorically-algebraic topology and its applications

This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...