Stone-Cech Compactification of Discrete Spaces (5.38.7-8)

James Munkres Topology, chapter 5.38, exercises 7-8:

7. Let $X$ be a discrete space.
(a) Show that if $A⊆X$, then $\overline{A}∩\overline{X-A}=ø$ where their closures are taken in $β(X)$.
(b) Show that if $U⊆β(X)$ is open, then $\overline{U}$ is open.
(c) Show that $X$ is totally disconnected.

8. Show the cardinality of $β(ℕ)$ is at least as great as $I^I$ where $I=[0,1]$.

Proof: 7. (a) Define a function $f : X→ℝ$ by $f(A)=1$ and $f(X-A)=0$. Letting $F : β(X)→ℝ$ extend $f$, we see $F^{-1}(1)$ and $F^{-1}(0)$ are disjoint closed sets in $β(X)$ containing $A$ and $X-A$ respectively, so these latters' closures are disjoint.

(b) Note $\overline{U∩X}∪\overline{X-U∩X}$ is the whole space $β(X)$ since its complement is an open set not intersecting $X$, so by part (a) $\overline{U∩X}$ is open. Now evidently $\overline{U∩X}⊆\overline{U}$, but also $\overline{U}⊆\overline{U∩X}$ since if there exists $x∈\overline{U}$ with a neighborhood $V$ disjoint from $U∩X$, let $y∈V∩U$ and now $V∩U$ is a neighborhood of $y$ not intersecting $X$ hence $y∉\overline{X}$, a contradiction.

(c) Let $x,y∈β(X)$ be distinct. Since $β(X)$ is Hausdorff let $U$ be open such that $x∈U$ and $v∉\overline{U}$. Then by part (b) $\overline{U}∪β(X)-\overline{U}$ is a separation of $β(X)$ disconnecting $x$ from $y$.

8. As we've seen (cf. 5.31.16a), $ℝ^I≅(0,1)^I$ has a countable dense subset, so $\overline{(0,1)^I}=[0,1]^I$ does as well, call it $S$. Letting $f : ℕ→S⊆[0,1]^I$ be a surjection that is automatically continuous, we obtain a map of $β(ℕ)$ into $I^I$ containing $S$, and since images of compact sets are compact hence closed in a Hausdorff space, the map is a surjection and the claim is proven.$~\square$