$ \def\bbN{{\mathbb N}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \def\calT{{\mathcal T}} \def\Lim{{\operatorname{Lim}}} $

(9)

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Homework Assignment #2

Read sections 14 through 17 in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread sections 18 through 22, just to get a feel for the future.

Solve and submit the following problems. In Munkres' book, problems 4, 5, and 8 on page 92, problems 4 and 7 on page 100, and, for extra credit, the following challenge problem:

Challenge Problem. Let $X$ and $Y$ be topological spaces and let $A\subset X$ and $B\subset Y$ be subsets thereof. Using only the definitions in terms of continuity of certain functions, show that the topology induced on $A\times B$ as a subset of the product $X\times Y$ is equal to the topology induced on it as a product of subsets of $X$ and of $Y$. You are allowed to use the fact that two topologies ${\mathcal T}_1$ and ${\mathcal T}_2$ on some set $W$ are equal if and only if the identity map regarded as a map from $(W, {\mathcal T}_1)$ to $(W, {\mathcal T}_2)$ is a homeomorphism. Words like "open sets" and "basis for a topology" are not allowed in your proof.

Note on Question 8, Page 92. One should think that "describe" for verbal things is like "simplify" for formula-things. The topologies in question were given by a verbal description; the content of the question is that you should be giving a simpler one, and the best is if it is of the form "the topology in question is the trivial topology", or something like that. Note that the resulting topology may also depend on the direction of the line L, so you may wish to divide your answer into parts depending on that direction.

Due date. This assignment is due at the end of class on Thursday, September 27, 2018.