\( \def\bbN{{\mathbb N}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \def\calT{{\mathcal T}} \def\Lim{{\operatorname{Lim}}} \)
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Homework Assignment #8

Warning. An unusually difficult assignment once again; constructive cooperation is highly encouraged! (Though as always, you'll lose if you don't struggle some on your own).

Post. If you have an excellent solution set for a past assignment, I'll be happy to post it as explained at About.html under "Solution Sets".

Read sections 30-35 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 43 and 45, just to get a feel for the future.

Solve and submit your solution of problem 2 on page 223 of Munkres' book.

In addition, solve the following multi-part problem. Some parts of this problem will be discussed and further hints will be given in the tutorials on Monday November 26. To some extent, like previous discussions we've had in class about "products" and "subspaces" and "quotients", what one learns in doing this question lies in the likes of algebra and category theory, even though neither is mentioned within the question. For the question is about "universal properties" and "universal constructions" and how to manage them.

Before we start, some philosophy: It is often useful to know that a space $X$ is a subset of a compact space $C$, and we often "don't care" about parts of $C$ that are "far" from $X$, so we may as well assume that $\overline{X}=C$. Such a space $C$ is called a "compactification" of $X$. The problem below shows that there is a (unique) "monster" compactification that is bigger than any other one.

Problem. Let $X$ be a completely regular space ($T_{3.5}$). A pair $(\beta X,\iota\colon X\to\beta X)$ is called "a Stone-Čech compactification" of $X$ if the following two properties hold:

  1. $\beta X$ is compact Hausdorff and $\iota$ is continuous.
  2. If $Y$ is any compact Hausdorff space and a continuous function $f\colon X\to Y$ is given, then there is a unique continuous function $\tilde{f}\colon\beta X\to Y$ such that $\tilde{f}\circ\iota=f$ ($\tilde{f}$ is called "the extension of $f$").

Questions (it will be announced on this page after the tutorials on November 26 which parts are to be submitted submit your solutions for parts 1, 2, 6, 7, and 8):

  1. Show that if $(\beta X,\iota)$ is as above, then $\iota$ is injective. (Hint: real valued functions separate the points of $X$.)
  2. Show that if $(\beta X,\iota)$ is as above, then $\iota$ is an embedding; namely, that $\iota$ induces a homeomorphism of $X$ with $\iota(X)$. (Hint: it is useful to remember that the topology of $X$ is generated by sets of the form $[f>0]$.)
  3. Show that if $(\beta X,\iota)$ is as above, then $\iota(X)$ is dense in $\beta X$; namely that $\overline{\iota(X)}=\beta X$. (Hint: contemplate the word "unique", which appeared in the definition.)
  4. Show that any two Stone-Čech compactifications of $X$ are equivalent. Namely, if $(\beta_1X,\iota_1)$ and $(\beta_2X,\iota_2)$ both satisfy the conditions above, then there exists a homeomorphism $\psi\colon\beta_1X\to\beta_2X$ such that $\iota_2=\psi\circ\iota_1$.
  5. Let $X$ be $T_{3.5}$, let $C_X$ be the set of continuous functions from $X$ to $I=[0,1]$, and let $\phi\colon X\to I^{C_X}$ be the embedding of $X$ into a cube as discussed in class (namely, $\phi(x)_f=f(x)$). Let $\beta_0X$ be $\overline{\phi(X)}$ and let $\iota_0$ be $\phi$ with its target space restricted to $\beta_0X$. Prove that if $f\colon X\to I$ is continuous then it has a unique "extension" $\tilde{f}\colon\beta_0X\to I$ such that $\tilde{f}\circ\iota_0=f$.
  6. Prove that if $Y$ is compact Hausdorff then $\iota_0\colon Y\to\beta_0Y$ (defined in the same way, with $Y$ replacing $X$) is a homeomorphism.
  7. Prove that $(\beta_0X,\iota_0)$ is a Stone-Čech compactification of $X$. (Hint: instead of extending maps into $Y$, it is easier to extend them into $\beta_0Y$, "coordinate by coordinate".)
  8. Prove that if $C$ is any compactification of $X$, then there is a surjection $\beta X\to C$.

Due date. This assignment is due at the end of class on Thursday, November 29, 2018. If you can, please use the Homework Submission Cover Page to help with faster returns and to help with privacy.

A Painful Reminder. Right after our final exam agents of the Evil Galactic Empire will lock all the students of this class in separate sound proof, electromagnetically sealed, neutrino hardened, and gravitational wave resistant rooms in the dark, cold lower basement of Sidney Smith Hall. In the rooms they will place identical countable sequences of numbered boxes, each one containing a real number (the same sequence of real numbers in each room). By the day after, each student must open all but one of their boxes in the order of their liking, and guess the number in the remaining box. If more than one student will guess wrong [oh no, redacted].
Do Something! You must devise a survival strategy before the final or else [too sad to write].

("Saw Omega" from Alfonso Gracia-Saz from Mira Bernstein from Vigorous Handwaving [spoilers inside]. Deadly serious.)