On the Algebraic Representation of Semicontinuity
نویسنده
چکیده
Abstract: The concepts of upper and lower semicontinuity in pointfree topology were introduced and first studied by Li and Wang in 1997. However Li and Wang’s treatment does not faithfully reflect the original classical notion. In this note, we present algebraic descriptions of upper and lower semicontinuous real functions, in terms of frame homomorphisms, that suggest the right alternative to the definitions of Li and Wang. This fixes the discrepancy between the classical and the pointfree notions and turns out to be the appropriate notion that makes the Katětov-Tong Theorem provable in the pointfree context without any restrictions.
منابع مشابه
Spectrum of plane curves via knot theory
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 ...
متن کاملLower semicontinuity for parametric set-valued vector equilibrium-like problems
A concept of weak $f$-property for a set-valued mapping is introduced, and then under some suitable assumptions, which do not involve any information about the solution set, the lower semicontinuity of the solution mapping to the parametric set-valued vector equilibrium-like problems are derived by using a density result and scalarization method, where the constraint set $K$...
متن کاملOn nuclei of sup-$Sigma$-algebras
In this paper, algebraic investigations on sup-$Sigma$-algebras are presented. A representation theorem for sup-$Sigma$-algebras in terms of nuclei and quotients is obtained. Consequently, the relationship between the congruence lattice of a sup-$Sigma$-algebra and the lattice of its nuclei is fully developed.
متن کاملNon-upper-semicontinuity of Algebraic Dimension for Families of Compact Complex Manifolds
In this note we show that in a certain subfamily of the Kuranishi family of any half Inoue surface the algebraic dimensions of the fibers jump downwards at special points of the parameter space showing that the upper semi-continuity of algebraic dimensions in any sense does not hold in general for families of compact non-Kähler manifolds. In the Kähler case, the upper semi-continuity always hol...
متن کاملA Class of compact operators on homogeneous spaces
Let $varpi$ be a representation of the homogeneous space $G/H$, where $G$ be a locally compact group and $H$ be a compact subgroup of $G$. For an admissible wavelet $zeta$ for $varpi$ and $psi in L^p(G/H), 1leq p <infty$, we determine a class of bounded compact operators which are related to continuous wavelet transforms on homogeneous spaces and they are called localization operators.
متن کامل