Local $\tau$-density of the sum and the superextension of topological spaces

In this paper, we study the density and the local density of the superextension of topological spaces. We prove that if $X_{\alpha}$ is a locally $\tau$-dense space for each $\alpha\in A$, then $X=\bigoplus \{X_{\alpha}: \alpha\in A\}$ is also a locally $\tau$-dense space. We also prove that for any infinite $T_{1}$-space, the inequality $ld(\lambda_{c}X)\le ld(X) $ is always valid.