We prove that an n-valued function that is lower semi-continuous is also upper semi-continuous and therefore continuous, that is, an n-valued map. Several conditions are shown to be equivalent to continuity for n-valued functions, in particular the continuity of a corresponding singlevalued function to a configuration space. Consequently, splitting is related to braid groups to prove a Splittin...

Douglas D. MOONEY, Thomas A. RICHMOND Western Kentucky University Bowling Green, KY 42101 USA The ideas of quotient maps, quotient spaces, and upper semicontinuous decompositions are extended to the setting of ordered topological spaces. These tools are used to investigate the semilattice of ordered compactifications and to construct ordered compactifications with o-totally disconnected and o-z...

We prove two relative local variational principles of topological pressure functions P (T,F ,U , y) and P (T,F ,U|Y ) for a given factor map π, an open cover U and a subadditive sequence of real-valued continuous functions F . By proving the upper semi-continuity and affinity of the entropy maps h{·}(T,U | Y ) and h+{·}(T,U | Y ) on the space of all invariant Borel probability measures, we show...

In the paper, the notion of a generalized convexity was defined and studied from the view-point of the selection and approximation theory of set-valued maps. We study the simultaneous existence of continuous relative selections and graph-approximations of lower semicontinuous and upper semicontinuous set-valued maps with α-convex values having nonempty intersection.

In this paper, we use topological methods to study various semicontinuity properties of the local spectrum of singular points of algebraic plane curves and spectrum at infinity of polynomial maps in two variables. Using the Seifert form, the Tristram–Levine signatures of links, and the associated Murasugi-type inequalities, we reprove (in a slightly weaker form) a result obtained by Steenbrink ...

Suppose that (X,A) is a measurable space and Y is a metrizable, Souslin space. Let Au denote the universal completion of A. For x ∈ X, let f(x, ·) be the lower semicontinuous hull of f(x, ·). If f : X × Y → R is (Au ⊗ B(Y ),B(R))-measurable, then f is (Au ⊗ B(Y ),B(R))-measurable.

We generalize a major portion of the classical theory of Cand C embedded subspaces to pointfree topology, where the corresponding notions are frame Cand C -quotients. The central results characterize these quotients and generalize Urysohn’s Extension Theorem, among others. The proofs require calculations in CL, the archimedean f -ring of frame maps from the topology of the reals into the frame ...

An algorithm for detecting periodic orbits in chaotic systems [Phys. Rev. E, 60 (1999), pp. 6172–6175], which combines the set of stabilising transformations proposed by Schmelcher and Diakonos [Phys. Rev. Lett., 78 (1997), pp. 4733–4736] with a modified semi-implicit Euler iterative scheme and seeding with periodic orbits of neighbouring periods, has been shown to be highly efficient when appl...