Approximation by continuous functions and its applications

Anton E. Lipin, Alexander V. Osipov

Published 2024-03-06Version 1

We prove that for every normal topological space $X$ and any function $f: X \to \mathbb{R}$ there is a continuous function $g : X \to \mathbb{R}$ such that $$|f(x) - g(x)| \leq \frac{1}{2} \sup\limits_{p \in X} \inf\limits_{O(p)} \sup\limits_{a,b \in O(p)} |f(a) - f(b)|$$ for all $x \in X$. As an application of this result we prove the following statements to types of tightness in a space $Q_p(X, \mathbb{R})$ of all quasicontinuous real-valued functions with the topology $\tau_p$ of pointwise convergence: the countability of tightness (fan-tightness, strong fan-tightness) at a point $f$ of space $Q_p(X, \mathbb{R})$ implies the countability of tightness (fan-tightness, strong fan-tightness) of space $Q_p(X,Y)$ of all quasicontinuous functions from $X$ into any non-one-point metrizable space $Y$. This result is the answer to the open question in the class of metrizable spaces.