On pointwise approximation of arbitrary functions by countable families of continuous functions
The following problem is considered. Given a real-valued function f defined on a topological space X, when can one find a countable familyfn:n∈ω of continuous real-valued functions on X that approximates f on finite subsets of X? That is, for any finite set F⊂X and every real number e>0 one can choosen∈ω such that ∥f(x)−fn(x)∥<e for everyx∈F. It will be shown that the problem has a positive solution if and only if X splits. A space X is said to split if, for any A⊂X, there exists a continuous mapfA:X→Rω such that A=fA−1(A). Splitting spaces will be studied systematically.
The purpose of this paper is to study a new class of framed manifolds. Such manifolds are called almost -cosymplectic f -manifolds. For some special cases of and s, one obtains (almost) -cosymplectic, (almost) C-manifolds, and (almost) Kenmotsu f -manifolds. Moreover, several tensor conditions are studied. We conclude our results with a general example on -cosymplectic f -manifolds.