Two Examples of Q-topologies

Rade Živaljević

A pair $(Y,\tau)$, where $Y$ is an internal set, whereas $\tau$ is a topology (usually external) on $Y$, is called a $^*$-topological space if $\tau$ has an internal base. The main example is $(^*X,\overline\tau)$ where $(X,\tau)$ is a standard topological space and $\overline\tau$ the topology generated by $^*\tau$. This is the so called $Q$-topology on $^*X$ induced by $(X,\tau)$, a notion introduced by A. Robinson in [4]. This note contains negative answers to some questions of R. W. Button, [1], who asked whether the following implications $$ \align &(^*X,\overline\tau)\enskip\text{normal}\enskip\Rightarrow (X,\tau) \enskip\text{normal}\\ &(X,\tau)\enskip\text{scattered}\enskip\Rightarrow (^*X,\overline\tau) \enskip\text{scattered} \endalign $$ hold in some enlargement.