\(
\def\bbN{{\mathbb N}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\calT{{\mathcal T}}
\def\Lim{{\operatorname{Lim}}}
\)
Homework Assignment #4
Read sections 19 through 21 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 22 through 24, just to get a feel for the future.
Solve following problems, though submit only the
underlined ones. In Munkres' book, problems 2, 3,
4, 5, 6, and 8 on pages 126-128.
In addition, solve and submit the following problem: Show that the following three spaces are homeomorphic:
- $\{0,1\}^\omega_{\text{dict}}$, the set of infinite sequences of 0's and 1's, in the dictionary order topology.
- $\{0,1\}^\omega_{\text{prod}}$, the same set, in the product
(cylinder) topology. (Each factor, the 2-element set $\{0,1\}$, is taken
with the discrete topology).
- The Cantor set $C$. (Taken with the topology it inherits from $\bbR$).
Due date. This assignment is due at the end of class on Thursday,
October 11, 2018.
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