# Exercise 7.59#

For 1., you can see $$U = [-∞,0) ∪ (0,∞]$$. Then the complement $$U^C = [∞,0] ∪ [0,∞]$$ is clearly $$[0]$$ or $$\{0\}$$ (a point). As expected, the complement of an open set is a closed set.

The Interior (topology) of [0] is not “(0)” because an open set (an “open line”) on the real line is defined to include all points within ε of 0, where ε is non-zero (see also Ball (mathematics)). It may be best not to even think of “(0)” or “(0,0)” as an open set; these use interval notation but without two or without two different numbers to define the interval. This contrasts with the acceptable notation [0] or [0,0], but so be it. So the largest open set contained in [0] is ∅.

If we were considering the discrete topology on ℝ (rather than the standard topology), then we would consider the point {0} both open and closed.

For 2. the complement of ∅ is the full set. The interior of the full set is the full set.

The primary lesson here is that the pseudocomplement of the pseudocomplement of $$U$$ is not always equal to $$U$$.

We’ve also selected a $$U$$ that is not a Regular open set. That is, it is not equal to the interior of its closure.

For 3., yes. For 4., no. From Heyting algebra:

Although the negation operation ¬a is not part of the definition, it is definable as a → 0. The intuitive content of ¬a is the proposition that to assume a would lead to a contradiction. The definition implies that a ∧ ¬a = 0. It can further be shown that a ≤ ¬¬a, although the converse, ¬¬aa, is not true in general, that is, double negation elimination does not hold in general in a Heyting algebra.