Set-theoretical entropies of weighted generalized shifts

In this paper for a finite field $F$, a nonempty set $\Gamma$, a self--map $\varphi:\Gamma\to\Gamma$ and a weight vector $\mathfrak{w}\in F^\Gamma$, we show that the set--theoretical entropy of the weighted generalized shift $\sigma_{\varphi,\mathfrak{w}}:F^\Gamma\to F^\Gamma$ is either zero or $+\infty$, moreover it is equal to zero if and only if $\sigma_{\varphi,\mathfrak{w}}$ is quasi--periodic. On the other hand after characterizing all conditions under which $\sigma_{\varphi,\mathfrak{w}}:F^\Gamma\to F^\Gamma$ is of finite fibre, we show that the cotravariant set--theoretical entropy of the finite fibre $\sigma_{\varphi,\mathfrak{w}}:F^\Gamma\to F^\Gamma$ depends only on $\varphi$ and $\rm{supp}(\mathfrak{w})$. In final sections we study the restriction of $\sigma_{\varphi,\mathfrak{w}}$ to the direct sum $\mathop{\bigoplus}\limits_{\Gamma}F$

Uniformizable functional Alexandroff spaces

In the following text we show that the Alexandroff space $X$ is uniformizable if and only if the collection of all smallest neighbourhoods is a partition of $X$. Moreover the Alexandroff space $X$ is uniformizable and functional Alexandroff ($k-$primal) if and only if the collection of all smallest neighbourhoods is a partition of $X$ into its finite subsets.Comment: 8 page