On I-convergence of sequences of functions

Abstract

Let $X$ be a topological space, $Y$ a uniform space and $\mathcal{I}$ an admissible ideal on the set $\mathbb{N}$ of natural numbers. In this paper, we mainly study the conditions that be added to pointwise $\mathcal{I}$-convergence of a sequence of (continuous) functions in $Y^X$ to preserve the continuity of the $\mathcal{I}$-limit function. Ideal versions of weakly exhaustive, semi exhaustive, semi uniform convergence, $\alpha$-convergence and semi-$\alpha$ convergence of sequences of functions are introduced. Their relationships are clarified. Assume that a sequence of functions $\{f_n\}_{n \in \mathbb{N}}$ pointwise $\mathcal{I}$-converges to $f$, we prove that:

(a) $f$ is continuous if and only if the sequence $\{f_n\}_{n \in \mathbb{N}}$ is $\mathcal{I}$-weakly exhaustive.

(b) If the sequence $\{f_n\}_{n \in \mathbb{N}}$ is $\mathcal{I}$-semi exhaustive, then $f$ is continuous.

(c) If the sequence $\{f_n\}_{n \in \mathbb{N}}$ $\mathcal{I}$-semi uniform converges to $f$ and $f_n$
is continuous for every $n \in \mathbb{N}$, then $f$ is continuous.

(d) If $\mathcal{I}$ is ``good" and $X$ is first countable, then $\{f_n\}_{n \in \mathbb{N}}$
is $\mathcal{I}$-$\alpha$ convergent to $f$ if and only if
$\{f_n\}_{n \in \mathbb{N}}$ is $\mathcal{I}$-exhaustive.

(e) If the sequence $\{f_n\}_{n \in \mathbb{N}}$ $\mathcal{I}$-semi-$\alpha$ converges to $f$, then $f$ is continuous.