Due October 2 in class

Section 20 problems 8a, 10
Section 21 problems 4, 6
Section 22 problems 2, 3, 4

then

A. Let \(t\) denote the taxicab metric on \(\mathbb{R}^n\), given by \(t((x_1,\ldots, x_n),(y_1,\ldots,y_n)) = \sum_{i=1}^n \left|x_i – y_i\right| \). Show that \(t\) and the euclidean metric induce the same topology on \(\mathbb{R}^n\).

B. Define \(\sim\) on \(\mathbb{R}\) by \(x\sim y\) whenever \(x-y \in\mathbb{Z}\). The quotient \(\mathbb{R}/\sim\)is also denoted as \(\mathbb{R}/\mathbb{Z}\). Does \(\mathbb{R}/\mathbb{Z}\) look like any topological space that we are familiar with? Justify your answer. Secondly, define a partition \(\mathbb{R}^\ast\) consisting of the singletons \(\{x\}\) if \(x\not\in \mathbb{Z}\) and the set \(\mathbb{Z}\). Do you think the quotient space \(\mathbb{R}^\ast\) is homeomorphic to \(\mathbb{R}/\mathbb{Z}\) (Do your best to answer this last question…it will be difficult to prove this rigorously. Appeal to your intuition here)?

C. Recall from class the line with two origins. It inherits a quotient topology from \(\mathbb{R}\times 1 \sqcup \mathbb{R}\times -1 \subset \mathbb{R}^2\), and it inherits the subspace topology from \(\mathbb{R}^2\). Compare and contrast the two topologies on the line with two origins.

D. Is a quotient of a Hausdorff space necessarily Hausdorff? Prove it or provide a counterexample.