91Ó°ÊÓ

Determine the set of interior points, accumulation points, isolated points, and boundary points for each of the following sets: (a) \(\\{1,1 / 2,1 / 3,1 / 4,1 / 5, \ldots\\}\) (b) \(\\{0\\} \cup\\{1,1 / 2,1 / 3,1 / 4,1 / 5, \ldots\\}\) (c) \((0,1) \cup(1,2) \cup(2,3) \cup(3,4) \cdots \cup(n, n+1) \cup \ldots\) (d) \((1 / 2,1) \cup(1 / 4,1 / 2) \cup(1 / 8,1 / 4) \cup(1 / 16,1 / 8) \cup \ldots\) (e) \(\\{x:|x-\pi|<1\\}\) (f) \(\left\\{x: x^{2}<2\right\\}\) (g) \(\mathbb{R} \backslash \mathbb{N}\) (h) \(\mathbb{R} \backslash \mathbb{Q}\)

Cantor, in 1885, defined a set \(E\) to be dense-in-itself if \(E \subset E^{\prime} .\) Develop some facts about such sets. Include illustrative examples.

Show that no interior point of a set can be a boundary point, that it is possible for an accumulation point to be a boundary point, and that every isolated point must be a boundary point.

Show that every uncountable set of real numbers has a point of accumulation.

Let \(\mathcal{F}\) be a family of (nondegenerate) intervals; that is, each member of \(\mathcal{F}\) is an interval (open, closed or neither) but is not a single point. Suppose that any two intervals \(I\) and \(J\) in the family have no point in common. Show that the family \(\mathcal{F}\) can be arranged in a sequence \(I_{1}, I_{2}, \ldots\)

Express the open interval \((0,1)\) as a union of a sequence of closed sets. Can it also be expressed as an intersection of a sequence of closed sets?

Give an example of an open covering of the set \(Q\) of rational numbers that does not reduce to a finite subcover.

If \(A\) and \(B\) are both open or both closed, what can you say about the sets \(A \backslash B\) and \(B \backslash A ?\)

Show that there is no set which has the interval \((0,1)\) as its set of accumulation points.