In the book "Integration Theory" (LNM315, K.Bichteler, p.65) a family $\mathcal{F}$ of real function is called full if it is close by pointwise convergence of dominated (by some element of $\mathcal{F}$) sequences. Given a general family $\mathcal{F}$ of real function its full-closure $\mathcal{F}^\delta$ (the minimal full-family containing $\mathcal{F}$) is calculate by transfinite induction: $\mathcal{F}^0:=\mathcal{F}$, and if $\mathcal{F}^a$ is defined for all $a<b$ let $\mathcal{F}^b$ the set of all pointwise limits of sequences with element's in $\cup_{a<b}\mathcal{F}^a$ (bounded by some element of this set).
If we consider uniform convergence (instead pointwise convergence) we can do in just one step.
For a topological space $X$ I consider the follow property:
$*)$ Given a convergent sequence of (real) functions $f_n$ such that $f_n\to f$ (pointwise) and a family of succession $g_{m,n}$ such that $g_{m,n }\to_m f_n$ (pointwise) then exist monotone function $\alpha,\beta: \mathbb{N}\to \mathbb{N}$ such that the sequence $h_j:=g_{\alpha(j), \beta(j)} $ converging (pointwise) to $f$.
If $(*)$ is true then $\mathcal{F}^\delta=\mathcal{F}^1$ i.e. we get $\mathcal{F}^\delta$ in just one step 's.
I'm looking for a example of a topological space $X$ such that the property $(*)$ isn't true.