1. If \((A_i)_{i \in I}\) is a collection of subsets of a set \(X\), prove the following: \(X \setminus \bigcup_{i \in I} A_i = \bigcap_{i \in I} X \setminus A_i.\) If \((B_j)_{j \in J}\) is another collection of subsets of \(X\), prove the following: \((\bigcup_{i \in I} A_i) \cap (\bigcup_{j \in J} B_j) = \bigcup_{i \in I} \bigcup_{j \in J} A_i \cap B_j\)

  2. Let \(A\) be a set of real numbers which is bounded, meaning there is a real number \(r\) that is an upper bound of \(A\) (i.e. \(a \le r\) for every \(a\) in \(A\)). We say that a real number \(s\) is a supremum of \(A\) (also known as least-upper-bound) if it is an upper bound of \(A\) and moreover if \(s'\) is another real number that bounds \(A\), then \(s \le s'\). Show that if a supremum of a set \(A\) exists, then it is unique. Now we confirm that supremums of bounded sets exist. This is known as the least-upper-bound property of the real numbers, which is a fundamental property of the reals. Let’s prove this assuming that Cauchy sequences converge. We define a sequence \((a_n)\) of elements of \(A\) and a sequence \((b_n)\) of upper bounds of \(A\) recursively as follows: let \(a_0\) be an arbitrary element of \(A\) and let \(b_0\) be an arbitrary upper bound of \(A\). In general if \(a_n\) and \(b_n\) are defined, check if the average of \(a_n\) and \(b_n\) is an upper bound of \(A\). If it is, let \(b_{n+1} = (a_n + b_n)/2\) and let \(a_{n+1} = a_n\). Otherwise, let \(a_{n+1} = (a_n + b_n)/2\) and \(b_{n+1} = b_n\). Show that these sequences are Cauchy and in fact converge to the same point, and that point is the supremum of \(A\).

  3. In-class exercise 1 from the first lecture 2025-01-06

  4. In-class exercise 2 from the first lecture 2025-01-06

  5. Manetti 3.3

  6. Manetti 3.4

  7. Show that the collection of intervals \((a, b)\) for rational \(a\) and \(b\) forms a basis for the standard topology. Show that the collection of half-open intervals \([a, b)\) for rational \(a\) and \(b\) does not form a basis for the lower-limit topology. Find a nice basis of the upper topology consisting of only countably-many sets.