A cardinal function $\varphi$ (or a property $𝒫$) is called $l$-invariant if for any Tychonoff spaces $X$ and $Y$ with $C_p\left(X\right)$ and $C_p\left(Y\right)$ linearly homeomorphic we have $\varphi \left(X\right)=\varphi \left(Y\right)$ (or the space $X$ has $𝒫$ ($\equiv X⊢𝒫$) iff $Y⊢𝒫$). We prove that the hereditary Lindelöf number is $l$-invariant as well as that there are models of $ZFC$ in which hereditary separability is $l$-invariant.

The fixed infinitely differentiable function $\rho \left(x\right)$ is such that $\left\{n\rho \right(nx\left)\right\}$ is a regular sequence converging to the Dirac delta function $\delta$. The function ${\delta }_{𝐧}\left(𝐱\right)$, with $𝐱=(x_1,\cdots ,x_m)$ is defined by $${\delta }_{𝐧}\left(𝐱\right)=n_1\rho \left(n_1{x}_1\right)\cdots n_m\rho \left(n_m{x}_m\right).$$ The product $f\circ g$ of two distributions $f$ and $g$ in ${𝒟}_m^{\text{'}}$ is the distribution $h$ defined by $$\underset{n_1\to \infty }{error}\cdots \underset{n_m\to \infty }{error}\langle f_{𝐧}{g}_{𝐧},\phi \rangle =\langle h,\phi \rangle ,$$ provided this neutrix limit exists for all $\phi \left(𝐱\right)={\phi }_1\left(x_1\right)\cdots {\phi }_m\left(x_m\right)$, where $f_{𝐧}=f*{\delta }_{𝐧}$ and $g_{𝐧}=g*{\delta }_{𝐧}$.

Let $𝒟$ denote a true dimension function, i.e., a dimension function such that $𝒟\left({ℝ}^n\right)=n$ for all $n$. For a space $X$, we denote the hyperspace consisting of all compact connected, non-empty subsets by $C\left(X\right)$. If $X$ is a countable infinite product of non-degenerate Peano continua, then the sequence ${\left({𝒟}_{\ge n}\left(C\left(X\right)\right)\right)}_{n=2}^{\infty }$ is $F_{\sigma }$-absorbing in $C\left(X\right)$. As a consequence, there is a homeomorphism $h:C\left(X\right)\to Q^{\infty }$ such that for all $n$, $h\left[\{A\in C\left(X\right):𝒟\left(A\right)\ge n+1\}\right]=B^n\times Q\times Q\times \cdots$, where $B$ denotes the pseudo boundary of the Hilbert cube $Q$. It follows that if $X$ is a countable infinite product of non-degenerate...