The lower dimension $\dim_L$ is the dual concept of Assouad dimension. As it fails to be monotonic, Fraser and Yu introduced modified $\dim_{ML}$ by making monotonic with simple formula $\dim_{ML} X=\sup\{\dim_L E: E\subset X\}$. our first result we prove that finitely stable in any metric space, answering a question Yu. We new, characterization for For space $X$ let $\mathcal{K}(X)$ denote non...

Baire category techniques are known to be a powerful tool in the investigation of the convex sets. Their use, which goes back to the fundamental contribution of Klee [17], has permitted to discover several interesting unexpected properties of convex sets (see Gruber [14], Schneider [23], Zamfirescu [25]). A survey of this area of research and additional bibliography can be found in [15, 27]. In...

Functions in Lploc[0,∞) where 1 ≤ p ≤ ∞ can be considered as inputs to linear systems. However, hysteresis operators of Preisach type have only been defined on much smaller space of regulated (or Baire) functions. In this paper, we re-define Play operators so that they are well defined for real valued measurable functions. We show that this definition coincides with the older definition for con...

In the theory of optimization an essential role is played by the differentiability of convex functions. In this paper we shall try to extend the results concerning differentiability to a larger class of functions called strongly α(·)-paraconvex. Let (X, ‖.‖) be a real Banach space. Let f(x) be a real valued strongly α(·)-paraconvex function defined on an open convex subset Ω ⊂ X , i.e. f ( tx+(...

Let X be a complete CAT(0) space with the geodesic extension property and Alexandrov curvature bounded below. It is shown that if C is a closed subset of X , then the set of points of X which have a unique nearest point in C is Gδ and of the second Baire category inX. If, in addition,C is bounded, then the set of points ofX which have a unique farthest point in C is dense in X. A proximity resu...

In [13] Moschovakis presents a topological model, over Baire space, for a system of intuitionistic analysis that includes relativized dependent choice, monotone bar induction (verified in [2, pp. 13-14]), Kripke’s schema, and the restriction of Brouwer’s principle for numbers to predicates without free choice-sequence parameters. Following Krol [12], one may describe this model roughly1 as foll...

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well-ordered), but for many other natural non-zerodimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called ∆ 0 α-reductions, and try to find for various natural topological spaces X the least ordinal αX such th...

We prove that for every singular cardinal μ of cofinality ω, the complete Boolean algebra compPμ(μ) contains a complete subalgebra which is isomorphic to the collapse algebra CompCol(ω1, μ0 ). Consequently, adding a generic filter to the quotient algebra Pμ(μ) = P(μ)/[μ] collapses μ0 to א1. Another corollary is that the Baire number of the space U(μ) of all uniform ultrafilters over μ is equal ...

It is proved that every function of finite Baire index on a separable metric space K is a D-function, i.e., a difference of bounded semi-continuous functions on K. In fact it is a strong D-function, meaning it can be approximated arbitrarily closely in D-norm, by simple D-functions. It is shown that if the n derived set of K is non-empty for all finite n, there exist D-functions on K which are ...