Let $S$ be an uncountably infinite set (mainly interested in case that $S$ has same cardinality as $\mathbb R$) and look at the set $F$ of functions $f\colon S \to \mathbb R$. I equip $F$ with the topology of pointwise convergence. It is not hard to show that $F$ is not FréchetUrysohn. On the other hand, define $F^f$ and $F^c \subseteq F$ to be the subsets of $F$ consisting of functions with finite, respectively countable, support. I'm trying to understand whether these spaces (equipped with the subspace topology from $F$, equivalently with the topology of pointwise convergence) are again FréchetUrysohn. My reason for expecting/hoping this is that my counterexample to the FréchetUrysohn property for $F$ makes essential use of functions with uncountable support; but this is of course not a proof. Thanks!
1 Answer
Yes, these spaces are FréchetUrysohn.
Consider $F^c$; the proof for $F^f$ is the same. Let $A \subset F^c$ and suppose $f \in \overline{A}$; since $F^c$ is a topological vector space (as is $F^f$) we can suppose without loss of generality that $f = 0$. Note this means that for every finite set $S_0 \subset S$ and every $\epsilon > 0$ we can find some $g \in A$ with $g(x) < \epsilon$ for every $x \in S_0$.
We construct inductively a sequence $f_n \in A$ which converges pointwise to $0$. Let $f_1$ be arbitrary and enumerate its support as $x_{1,1}, x_{1,2}, \dots$. Then choose $f_2$ such that $f_2(x_{1,1}) < 1/2$, and enumerate the support of $f_2$ as $x_{2,1}, x_{2,2}, \dots$. Then choose $f_3$ such that $f_3(x_{i,j}) < 1/3$ for $i,j = 1,2$, and continue in this way to produce a sequence $f_n$ with the property that $f_n(x_{i,j}) < 1/n$ for all $i,j = 1, \dots, n1$, where $\{x_{i,j}\}_{j=1}^\infty$ enumerates the support of $f_i$.
Now consider any arbitrary $x \in S$. If $x = x_{i,j}$ for some $i,j$, then letting $N = \max(i,j)$, we have $f_n(x) < 1/n$ for all $n \ge N$ and hence $f_n(x) \to 0$. Otherwise, $x$ is outside the support of all the $f_n$ and then $f_n(x) \to 0$ trivially.

$\begingroup$ Thank you so much! This has been driving me nuts... I missed the idea of including some of the support of $f_2$ in the set on which I make $f_3$ small. I’d somehow convinced myself it should really be false and was trying to construct counterexamples. I’ll sleep better now! $\endgroup$ Dec 17, 2020 at 22:17