Part 4 of completing S110

spaces/S000110/properties/P000022.md

@@ -2,10 +2,9 @@
 space: S000110
 property: P000022
 value: true
-refs:
-- doi: 10.1007/978-1-4612-6290-9_6
-  name: Counterexamples in Topology
 ---
 
-Asserted in the General Reference Chart for space #113 in
-{{doi:10.1007/978-1-4612-6290-9_6}}.
+For any continuous $f: X \to \mathbb{R}$, the definition of the topology ensures that $f(F) = \lim_{n \to F} f(n)$ for every
+$F \in M$, so $f$ is actually continuous as a function $\beta\omega \to \mathbb{R}$, where $\beta\omega$ is
+{S108}. Since {S108|P16},
+the range of $f$ is compact and thus bounded.

spaces/S000110/properties/P000032.md

@@ -0,0 +1,10 @@
+---
+space: S000110
+property: P000032
+value: false
+refs:
+- mathse: 5016607
+  name: Is the strong ultrafilter topology countably paracompact?
+---
+
+See {{mathse:5016607}}.

spaces/S000110/properties/P000033.md

@@ -0,0 +1,7 @@
+---
+space: S000110
+property: P000033
+value: true
+---
+
+Let $\{U_k\}_{k \in \omega}$ be a countable open cover of $X$. Since all subsets of $M$ are closed, we may first refine $\{U_k\}_{k \in \omega}$ to an open cover $\{V_k\}_{k \in \omega}$ where $M \cap V_k$ are pairwise disjoint. Then choose the further refinement $\mathscr{W} = \{\{n\}: n \in \omega\} \cup \{V_k \setminus \{0, \cdots, k\}: k \in \omega\}$. This refinement is point-finite. Indeed, for any $F \in M$, as $M \cap V_k$ are pairwise disjoint, $F$ belongs to exactly one $V_k$ so $F$ is contained in exactly one element of $\mathscr{W}$. For $n \in \omega$, it at most belongs to $\{n\}, V_0 \setminus \{0\}, \cdots, V_{n-1} \setminus \{0, \cdots, n-1\}$ and it cannot belong to any other element of $\mathscr{W}$. So, $\mathscr{W}$ is point-finite at $n$ as well.

spaces/S000110/properties/P000194.md

@@ -0,0 +1,7 @@
+---
+space: S000110
+property: P000194
+value: true
+---
+
+Let $\mathscr{U}$ be an open cover of $X$. For each $F \in M$, there exists $A_F \in F$ s.t. $A_F \cup \{F\}$ is contained in some element of $\mathscr{U}$. Thus, for each $k \in \omega$, we may consider the refinement $\mathscr{V}_k = \{\{n\}: n \in \omega\} \cup \{(A_F \setminus \{0, \cdots, k\}) \cup \{F\}: F \in M\}$. For any $F \in M$ and $k \in \omega$, $F$ is contained in exactly one element of $\mathscr{V}_k$, namely $(A_F \setminus \{0, \cdots, k\}) \cup \{F\}$. For any $n \in \omega$, $n$ is contained in exactly one element of $\mathscr{V}_n$, namely $\{n\}$. Hence, $\mathscr{V}_k$ form a $\theta$-sequence of open refinements of $\mathscr{U}$.