Trivial Spaces Research Articles

Dynamic topological S5

The topological semantics for modal logic interprets a standard modal propositional language in topological spaces rather than Kripke frames: the most general logic of topological spaces becomes S4. But other modal logics can be given a topological semantics by restricting attention to subclasses of topological spaces: in particular, S5 is logic of the class of almost discrete topological spaces, and also of trivial topological spaces. Dynamic Topological Logic (DTL) interprets a modal language enriched with two unary temporal connectives, next and henceforth. DTL interprets the extended language in dynamic topological systems: a DTS is a topological space together with a continuous function used to interpret the temporal connectives. In this paper, we axiomatize four conservative extensions of S5, and show them to be the logic of continuous functions on almost discrete spaces, of homeomorphisms on almost discrete spaces, of continuous functions on trivial spaces and of homeomorphisms on trivial spaces.

Fixed point theorems in homotopically trivial spaces

We introduce a new concept of generalized Knaster–Kuratowski–Mazurkiewicz (KKM) family of sets and related to this we obtain fixed point theorems and sections results in homotopically trivial spaces.

Formula omitted]-trivial spaces

Let X be a regular Hausdorff space and H a subcollection of the lattice F of all closed subsets of X, which is hereditary with respect to closed sets and stable under finite unions. The space X is called H -trivial if the co-compact topology on H coincides with the corresponding upper Kuratowski topology. We propose a description of H -trivial spaces in terms of open sieves, providing parallels and connections with Čech-complete and sieve-complete spaces. Answering a question of Nogura and Shakhmatov, we show that if X is of pointwise countable type, then X is locally compact if and only if X× Y is F -trivial (i.e., consonant) for every F -trivial space Y. A topological proof of the fact that the rationals are not Fin -trivial is given, where Fin is the lattice of finite sets. Other examples and results related to H -trivial spaces are discussed.

K3 surfaces and string duality

The primary purpose of these lecture notes is to explore the moduli space of type IIA, type IIB, and heterotic string compactified on a K3 surface. The main tool which is invoked is that of string duality. K3 surfaces provide a fascinating arena for string compactification as they are not trivial spaces but are sufficiently simple for one to be able to analyze most of their properties in detail. They also make an almost ubiquitous appearance in the common statements concerning string duality. We review the necessary facts concerning the classical geometry of K3 surfaces that will be needed and then we review old string theory on K3 surfaces in terms of conformal field theory. The type IIA string, the type IIB string, the E8 x E8 heterotic string, and Spin(32)/Z2 heterotic string on a K3 surface are then each analyzed in turn. The discussion is biased in favour of purely geometric notions concerning the K3 surface itself. These are an extended form of the notes from lectures given at TASI 96.