91Ó°ÊÓ

Let \(E\) be the set of endpoints of intervals complementary to the Cantor set \(K\). Prove that \(\bar{E}=K\).

Show that every subset of a set of measure zero also has measure zero.

Give an example of a sequence of nowhere dense sets whose union is not nowhere dense.

Show that the union of any sequence of first category sets is again a first category set.

Let \(G\) be a dense open subset of \(\mathbb{R}\) and let \(\left\\{\left(a_{k}, b_{k}\right)\right\\}\) be its set of component intervals. Prove that \(H=\mathbb{R} \backslash G\) is perfect if and only if no two of these intervals have common endpoints.

Which of the following statements are true? (a) Every subset of a nowhere dense set is nowhere dense. (b) If \(A\) is nowhere dense, then so too is \(A+c=\\{t+c: t \in A\\}\) for every number \(c\). (c) If \(A\) is nowhere dense, then so too is \(c A=\\{c t: t \in A\\}\) for every positive number \(c\). (d) If \(A\) is nowhere dense, then so too is \(A^{\prime}\), the set of derived points of \(A\). (e) A nowhere dense set can have no interior points. (f) A set that has no interior points must be nowhere dense. (g) Every point in a nowhere dense set must be isolated. (h) If every point in a set is isolated, then that set must be nowhere dense.

If \(E\) has measure zero, show that the translated set $$ E+\alpha=\\{x+\alpha: x \in E\\} $$ also has measure zero.

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous function. Assume that for every positive number \(\varepsilon\) the sequence \(\\{f(n \varepsilon)\\}\) converges to zero as \(n \rightarrow \infty\). Prove that $$ \lim _{x \rightarrow \infty} f(x)=0 $$

Prove that every set \(A\) is dense in its closure \(\bar{A}\).

If \(A\) is nowhere dense, what can you say about \(\mathbb{R} \backslash A ?\) If \(A\) is dense, what can you say about \(\mathbb{R} \backslash A ?\)