# Exercise 7.68#

Apply the above definition to the βperson $$π$$ is worried about news $$π‘$$β predicate from Example 7.65.

1. What open set is $$β(π‘: π). π(π ,π‘)$$ for a person $$π$$?

2. Does it have the semantic meaning youβd expect?

## Authorβs solution#

1. The formula says that $$β(π‘: π). π(π ,π‘)$$ βreturns the union $$V=\bigcup_{i}V_{i}$$ of all the open sets $$π_π$$ for which there exists some $$π‘_π β π(π_π)$$ satisfying $$π(π |_{π_i}, π‘_π) = π_π$$.β Substituting, this becomes βthe union of all time intervals $$π_π$$ for which there is some item $$π‘_π$$ in the news about which $$π$$ is worried throughout $$π_π$$.β In other words it is all the time that $$π$$ is worried about at least one thing in the news. Perhaps when $$π$$ is sleeping or concentrating on something, she is not worried about anything, in which case intervals of sleeping or concentrating would not be subsets of $$π$$. But if $$π$$ said βthereβs been such a string of bad news this past year, itβs like Iβm always worried about something!,β she is saying that itβs like $$π$$ = βthis past year.β

2. This seems like a good thing for βthere exists a piece of news that worries π β to mean: the news itself is allowed to change as long as the personβs worry remains. Someone might disagree and think that the predicate should mean βthere is one piece of news that worries π  throughout the whole interval π.β In that case, perhaps this person is working within a different topos, e.g. one where the site has fewer coverings. Indeed, it is the notion of covering that makes existential quantification work the way it does.

Itβs the union $$V=\bigcup_{i}V_{i}$$ of all the open sets $$π_π$$ for which there exists some $$π‘_π β π(π_π)$$ satisfying $$π(π |_{π_i}, π‘_π) = π_π$$ (to repeat the definition). In other words, itβs the times where the person $$s$$ is worried about at least one thing in the news (as in Example 7.65).