1. (Intermediate Value Theorem) Let \(X\) be a connected topological space and give \(\mathbb{R}\) the standard topology. Let \(f : X \rightarrow \mathbb{R}\) be a continuous function. Let \(a, b \in X\) and suppose \(r \in \mathbb{R}\) satisfies \(a < r < b\). Show that there is some \(c \in X\) with \(f(c) = r\). What if we didn’t want to assume \(X\) is connected? Is there a weaker assumption involving \(a, b\) we covered in class that would suffice?

  2. Show that if \(X\) and \(Y\) are two path-connected topological spaces then the product \(X \times Y\) is path-connected.

  3. Let \(X\) be a topological space. Define an equivalence relation \(\sim\) on \(X\) as follows: for any two points \(x, y \in X\), write \(x \sim y\) iff there is a connected set containing both \(x\) and \(y\). Prove that the equivalence classes \([x]_\sim\) are preciecely the connected components of \(X\).

  4. The following is proved in the textbook but I recommend you try to prove it yourself. A subset \(A \subseteq \mathbb{R}\) is called an interval if for every \(r < s \in A\) we have \((r, s) \subseteq A\). Show that the intervals of \(\mathbb{R}\) are precisely the sets of the form \((a, b)\), \([a, b)\), \((a, b]\), and \([a, b]\). Now give \(\mathbb{R}\) the standard topology. Show that a set \(A \subseteq \mathbb{R}\) is an interval iff it is connected iff it is path-connected.

  5. Recall the Baire space \(\mathbb{N}^{\mathbb{N}}\) from the second writtn assignment. Is it connected? If not, what are the connected components? What about for path-connectedness?

  6. Consider the function \(f : (-\infty, 0) \rightarrow \mathbb{R}\) defined by \(f(t) = sin(1/t)\). Draw the graph of this function. Let \(C = \{(0, y) \mid y \in [-1, 1]\}\). Show that \(graph(y) \cup C\) is connected but not path-connected. Prove this carefully!

  7. Manetti 4.17