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.

We consider a sequence of multi-bubble solutions $u_k$ of the following fourth order equation $${\Delta }^2{u}_k={\rho }_k\frac{h\left(x\right)e^{u_k}}{{\int }_{\Omega }h{e}^{u_k}}\text{in}\Omega ,u_k=\Delta u_k=0\text{on}\partial \Omega ,(*)$$ where $h$ is a $C^{2,\beta }$ positive function, $\Omega$ is a bounded and smooth domain in ${ℝ}^4$, and ${\rho }_k$ is a constant such that ${\rho }_k\le C$. We show that (after extracting a subsequence), ${lim}_{k\to +\infty }{\rho }_k=32{\sigma }_{3}m$ for some positive integer $m\ge 1$, where ${\sigma }_3$ is the area of the unit sphere in ${ℝ}^4$. Furthermore, we obtain the following sharp estimates for ${\rho }_k$: $$\begin{array}{cc}\hfill {\rho }_k-32{\sigma }_{3}m& =c_0\sum _{j=1}^m{ϵ}_{k,j}^2\left(\sum _{l\ne j}\Delta G_4(p_j,p_l)+\Delta R_4(p_j,p_j)+\frac{1}{32{\sigma }_3}\Delta logh\left(p_j\right)\right)\hfill \\ & +o\left(\sum _{j=1}^m{ϵ}_{k,j}^2\right)\hfill \end{array}$$ where $c_0\>0$, $log\frac{64}{{ϵ}_{k,j}^4}=\underset{x\in B_{\delta }\left(p_j\right)}{max}{u}_k\left(x\right)-log\left(\underset{\Omega }{\int }h{e}^{u_k}\right)$ and $u_k\to 32{\sigma }_3\sum _{j=1}^m{G}_4(·,p_j)$ in $C_{\mathrm{loc}}^4(\Omega \setminus \{p_1,...,p_m\})$. This yields a bound of solutions as...

A Banach space $X$ is called if for any cover $𝒰$ of $X$ by weakly open sets there is a finite subfamily $𝒱\subset 𝒰$ covering some ball of radius 1 centered at a point $x$ with $\parallel x\parallel \le r$. We prove that an infinite-dimensional separable Banach space $X$ is $\infty$-reflexive ($r$-reflexive for some $r\in \mathbb{N}$) if and only if each $\epsilon$-net for $X$ has an accumulation point (resp., contains a non-trivial convergent sequence) in the weak topology of $X$. We show that the quasireflexive James space $J$ is $r$-reflexive for no $r\in \mathbb{N}$. We do not know...

For a multivalued map $\varphi :Y⊸(X,\tau )$ between topological spaces, the upper semifinite topology $𝒜\left(\tau \right)$ on the power set $𝒜\left(X\right)=\{A\subset X:A\ne \varnothing \}$ is such that $\varphi$ is upper semicontinuous if and only if it is continuous when viewed as a singlevalued map $\varphi :Y\to \left(𝒜\right(X),𝒜(\tau \left)\right)$. In this paper, we seek a result like this from a reverse viewpoint, namely, given a set $X$ and a topology $\Gamma$ on $𝒜\left(X\right)$, we consider a natural topology $ℛ\left(\Gamma \right)$ on $X$, constructed from $\Gamma$ satisfying $ℛ\left(\Gamma \right)=\tau$ if $\Gamma =𝒜\left(\tau \right)$, and we give necessary and sufficient conditions to the upper semicontinuity of a multivalued...