The point of continuity property, neighbourhood assignments and filter convergences

Ahmed Bouziad. "The point of continuity property, neighbourhood assignments and filter convergences." Fundamenta Mathematicae 218.3 (2012): 225-242. <http://eudml.org/doc/282910>.

@article{AhmedBouziad2012,
abstract = {We show that for some large classes of topological spaces X and any metric space (Z,d), the point of continuity property of any function f: X → (Z,d) is equivalent to the following condition: (*) For every ε > 0, there is a neighbourhood assignment $(V_x)_\{x ∈ X\}$ of X such that d(f(x),f(y)) < ε whenever $(x,y) ∈ V_y × V_x$. We also give various descriptions of the filters ℱ on the integers ℕ for which (*) is satisfied by the ℱ-limit of any sequence of continuous functions from a topological space into a metric space.},
author = {Ahmed Bouziad},
journal = {Fundamenta Mathematicae},
keywords = {fragmentable function; point of continuity property; weakly separated function; separately continuous function; the first Baire class; filter convergence; -diagonalizable by -universal sets},
language = {eng},
number = {3},
pages = {225-242},
title = {The point of continuity property, neighbourhood assignments and filter convergences},
url = {http://eudml.org/doc/282910},
volume = {218},
year = {2012},
}

TY - JOUR
AU - Ahmed Bouziad
TI - The point of continuity property, neighbourhood assignments and filter convergences
JO - Fundamenta Mathematicae
PY - 2012
VL - 218
IS - 3
SP - 225
EP - 242
AB - We show that for some large classes of topological spaces X and any metric space (Z,d), the point of continuity property of any function f: X → (Z,d) is equivalent to the following condition: (*) For every ε > 0, there is a neighbourhood assignment $(V_x)_{x ∈ X}$ of X such that d(f(x),f(y)) < ε whenever $(x,y) ∈ V_y × V_x$. We also give various descriptions of the filters ℱ on the integers ℕ for which (*) is satisfied by the ℱ-limit of any sequence of continuous functions from a topological space into a metric space.
LA - eng
KW - fragmentable function; point of continuity property; weakly separated function; separately continuous function; the first Baire class; filter convergence; -diagonalizable by -universal sets
UR - http://eudml.org/doc/282910
ER -