31 research outputs found
Spaces with a -diagonal
A space has a -diagonal if has a
-directed compact cover. We show that any compact
space with a -diagonal is metrizable, hence any Tychonorff space
with a -diagonal is cosmic. These give a positive answer to Problem
4.2 and Problem 4.8 in \cite{COT11} raised by Cascales, Orihuela and Tkachuk
Spaces with a Finite Family of Basic Functions
A space X is finite dimensional, locally compact and separable metrizable if
and only if X has a finite basic family: continuous functions Phi_1,...,Phi_n
of X to the reals, R, such that for all continuous f from X to R there are
g_1,..., g_n in C(R) satisfying
f(x)=g_1(Phi_1(x))+g_2(Phi_2(x))+...+g_n(Phi_n(x)) for all x in X. This give
the complete solution to four problems on basic functions posed by Sternfeld,
as well as questions posed by Hattori and others
Hiblert's 13th Problem
The 13th Problem from Hilbert's famous list [16] asks whether every continuous function of three variables can be written as a superposition (in other words, composition) of continuous functions of two variables. Let Χ be a space. A family Φ ⊆ C(Χ) is said to be basic for Χ if each f in C(Χ) can be written as linear superposition for some functions from in Φ and some one-variable real functions. A family Ψ is elementary in dimension m if the family of maps generated by Ψ by addition is basic for Χ*…*Χ . Kolmogorov and Arnold [18, 4] showed that the closed unit interval has a finite elementary family in every dimension, thereby solving Hilbert's 13th Problem.Define a new cardinal invariant basic(Χ ) = min {|Φ|: Φ is a basic family for Χ}. It is established that a space has a finite basic family if and only if it is finite dimensional, locally compact and separable metrizable (or equivalently, homeomorphic to a closed subspace of Euclidean space).Such a space has dim(Χ) ≤ n if and only if basic(Χ) ≤ 2n+1. Separable metrizable spaces either have finite basic(Χ) or basic(Χ) equal to the continuum. The value of basic(K) for a compact space K is closely connected with the cofinality of the countable subsets of a basis B for K of minimal size ordered by set inclusion.It is proved that a space has a finite elementary family in every dimension m if and only if it is homeomorphic to a closed subspace of Euclidean space. It is further shown that there is a finite elementary family for the reals in each dimension m consisting of effectively computable functions, and effective procedures for representing any continuous function of m real variables as a superposition of these elementary functions and other univariate maps