S155/S157: Fortissimo spaces of size aleph_1 and aleph_2 (#1225)

properties/P000174.md

@@ -14,10 +14,13 @@ refs:
   name: "Name for topological spaces where 'every point has a local base wellordered by reverse inclusion'"
 ---
 
-Each point of the space has a neighborhood base well-ordered by reverse inclusion.
+Each point $x\in X$ has a neighborhood base well-ordered by reverse inclusion.
 
-Equivalently, each point of the space has a neighborhood base totally ordered by reverse inclusion.  The two are equivalent since a linear order admits a well-ordered cofinal subset.
+Equivalently, each point $x\in X$ has a neighborhood base totally ordered by reverse inclusion.  The two are equivalent since a linear order admits a well-ordered cofinal subset.
 
 Such spaces are called *well-based* in {{doi:10.1016/j.topol.2014.08.002}} and *lob-spaces* (lob = "linearly ordered (local) base") in {{mr:0540476}}.
 
+One says that $X$ is *well-based at the point $x$* if the conditions above hold at the specific point.
+So $X$ is well-based if it is well-based at every point.
+
 See also {{mo:202280}} and {{mo:322162}}.

spaces/S000022/README.md

@@ -12,10 +12,13 @@ refs:
     name: Fort space on Wikipedia
 ---
 
-Let $X=\mathbb{R}\cup\{\infty\}$.
-Every point of $\mathbb R$ is isolated and the open neighborhoods of the point $\infty$ are the cocountable subsets of $X$ containing that point.
+Let $X=\mathbb R$ and let $p$ be a distinguished point of $X$.
+Every point not equal to $p$ is isolated and the open neighborhoods of $p$ are the cocountable subsets of $X$ containing that point.
 
 This space is the one-point Lindelöfication of an uncountable discrete space.
 
+Compare with {S155} and {S157}.
+With a suitable assumption on $|\mathbb R|$ so that the cardinalities match, $X$ will be homeomorphic to one of them.
+
 Defined as counterexample #25 ("Fortissimo Space")
 in {{doi:10.1007/978-1-4612-6290-9}}.

spaces/S000022/properties/P000029.md

@@ -7,7 +7,5 @@ refs:
   name: Counterexamples in Topology
 ---
 
-$\{\{x\}\ |\ x \neq p\}$ is an uncountable antichain.
-
-Asserted in the General Reference Chart for space #25 in
-{{doi:10.1007/978-1-4612-6290-9_6}}.
+The singletons $\{x\}$ for $x\ne p$ are open and pairwise disjoint.
+And there are uncountably many of them.

spaces/S000022/properties/P000147.md

@@ -7,5 +7,5 @@ refs:
   name: Counterexamples in Topology
 ---
 
-The point $\infty$ is a P-point, since its neighborhoods are the cocountable subsets of $X$ containing the point and a countable intersection of them is still such a neighborhood.
+The point $p$ is a P-point, since its neighborhoods are the cocountable subsets of $X$ containing the point and a countable intersection of them is still such a neighborhood.
 And every isolated point is also a P-point.

spaces/S000022/properties/P000151.md

@@ -7,4 +7,4 @@ refs:
   name: What kinds of selection principles hold for Fortissimo space?
 ---
 
-See {{mathse:4727833}}.
+See {{mathse:4727833}}.

spaces/S000022/properties/P000172.md

@@ -0,0 +1,10 @@
+---
+space: S000022
+property: P000172
+value: true
+refs:
+- mathse: 4833761
+  name: LOTS and radial properties for generalized Fort/Fortissimo spaces
+---
+
+See Proposition 1 in {{mathse:4833761}}, with $\kappa=\omega_1$.

spaces/S000022/properties/P000203.md

@@ -4,4 +4,4 @@ property: P000203
 value: true
 ---
 
-All points except $\infty$ are isolated.
+All points except $p$ are isolated.

spaces/S000155/README.md

@@ -0,0 +1,20 @@
+---
+uid: S000155
+name: Fortissimo space of size $\aleph_1$
+refs:
+  - doi: 10.1007/978-1-4612-6290-9 
+    name: Counterexamples in Topology
+  - wikipedia: Fort_space
+    name: Fort space on Wikipedia
+---
+
+Let $X$ be a set of cardinality $\aleph_1$ with a distinguished point $p\in X$.
+Every point not equal to $p$ is isolated and the open neighborhoods of $p$ are the cocountable subsets of $X$ containing that point.
+
+This space is the one-point Lindelöfication of a discrete space of size $\aleph_1$.
+
+If (CH) $2^{\aleph_0}=\aleph_1$ holds, this space is homeomorphic to {S22}.
+Compare also with {S157}.
+
+For a general $X$ with unspecified uncountable cardinality,
+see counterexample #25 ("Fortissimo Space") in {{doi:10.1007/978-1-4612-6290-9}}.

spaces/S000155/properties/P000114.md

@@ -0,0 +1,7 @@
+---
+space: S000155
+property: P000114
+value: true
+---
+
+By construction.

spaces/S000155/properties/P000133.md

@@ -0,0 +1,10 @@
+---
+space: S000155
+property: P000133
+value: true
+refs:
+- mathse: 4833761
+  name: LOTS and radial properties for generalized Fort/Fortissimo spaces
+---
+
+See Proposition 2 in {{mathse:4833761}}, where $X$ is $F_{\omega_1,\omega_1}$.

spaces/S000155/properties/P000147.md

@@ -0,0 +1,8 @@
+---
+space: S000155
+property: P000147
+value: true
+---
+
+The point $p$ is a P-point, since its neighborhoods are the cocountable subsets of $X$ containing the point and a countable intersection of them is still such a neighborhood.
+And every isolated point is also a P-point.

spaces/S000155/properties/P000151.md

@@ -0,0 +1,10 @@
+---
+space: S000155
+property: P000151
+value: true
+refs:
+- mathse: 4727833
+  name: What kinds of selection principles hold for Fortissimo space?
+---
+
+See {{mathse:4727833}}.

spaces/S000155/properties/P000174.md

@@ -0,0 +1,18 @@
+---
+space: S000155
+property: P000174
+value: true
+refs:
+- mathse: 4833761
+  name: LOTS and radial properties for generalized Fort/Fortissimo spaces
+---
+
+Consider the ordered set $Y=((\omega_1\times\mathbb Z)\cup\{\infty\},<)$,
+where the product $\omega_1\times\mathbb Z$ has the lexicographic order
+and the element $\infty$ is larger than all the others.
+The space $X$ is homeomorphic to $Y$ with its corresponding order topology
+(see the proof of Proposition 2 in {{mathse:4833761}}).
+
+$X$ is well-based at the point $\infty$, since the intervals $(\alpha,\infty]$
+for $\alpha<\infty$ form a neighborhood base totally ordered by inclusion.
+And $X$ is trivially well-based at each of the other (isolated) points.

spaces/S000155/properties/P000203.md

@@ -0,0 +1,7 @@
+---
+space: S000155
+property: P000203
+value: true
+---
+
+All points except $p$ are isolated.

spaces/S000157/README.md

@@ -0,0 +1,20 @@
+---
+uid: S000157
+name: Fortissimo space of size $\aleph_2$
+refs:
+  - doi: 10.1007/978-1-4612-6290-9 
+    name: Counterexamples in Topology
+  - wikipedia: Fort_space
+    name: Fort space on Wikipedia
+---
+
+Let $X$ be a set of cardinality $\aleph_2$ with a distinguished point $p\in X$.
+Every point not equal to $p$ is isolated and the open neighborhoods of $p$ are the cocountable subsets of $X$ containing that point.
+
+This space is the one-point Lindelöfication of a discrete space of size $\aleph_2$.
+
+If $2^{\aleph_0}=\aleph_2$ holds, this space is homeomorphic to {S22}.
+Compare also with {S155}.
+
+For a general $X$ with unspecified uncountable cardinality,
+see counterexample #25 ("Fortissimo Space") in {{doi:10.1007/978-1-4612-6290-9}}.

spaces/S000157/properties/P000002.md

@@ -0,0 +1,7 @@
+---
+space: S000157
+property: P000002
+value: true
+---
+
+By inspection, every singleton is closed.

spaces/S000157/properties/P000029.md

@@ -0,0 +1,8 @@
+---
+space: S000157
+property: P000029
+value: false
+---
+
+The singletons $\{x\}$ for $x\ne p$ are open and pairwise disjoint.
+And there are uncountably many of them.

spaces/S000157/properties/P000059.md

@@ -0,0 +1,8 @@
+---
+space: S000157
+property: P000059
+value: true
+---
+
+$\aleph_1\le\mathfrak c$.
+Hence $\aleph_2=\aleph_1^+\le\mathfrak c^+\le 2^{\mathfrak c}$.

spaces/S000157/properties/P000114.md

@@ -0,0 +1,7 @@
+---
+space: S000157
+property: P000114
+value: false
+---
+
+$|X|=\aleph_2\ne\aleph_1$.

spaces/S000157/properties/P000147.md

@@ -0,0 +1,8 @@
+---
+space: S000157
+property: P000147
+value: true
+---
+
+The point $p$ is a P-point, since its neighborhoods are the cocountable subsets of $X$ containing the point and a countable intersection of them is still such a neighborhood.
+And every isolated point is also a P-point.

spaces/S000157/properties/P000151.md

@@ -0,0 +1,10 @@
+---
+space: S000157
+property: P000151
+value: true
+refs:
+- mathse: 4727833
+  name: What kinds of selection principles hold for Fortissimo space?
+---
+
+See {{mathse:4727833}}.

spaces/S000157/properties/P000154.md

@@ -0,0 +1,10 @@
+---
+space: S000157
+property: P000154
+value: false
+refs:
+- mathse: 4833761
+  name: LOTS and radial properties for generalized Fort/Fortissimo spaces
+---
+
+See Proposition 4 in {{mathse:4833761}}, where $X$ is $F_{\omega_2,\omega_1}$.

spaces/S000157/properties/P000172.md

@@ -0,0 +1,10 @@
+---
+space: S000157
+property: P000172
+value: true
+refs:
+- mathse: 4833761
+  name: LOTS and radial properties for generalized Fort/Fortissimo spaces
+---
+
+See Proposition 1 in {{mathse:4833761}}, where $X$ is $F_{\omega_2,\omega_1}$.

spaces/S000157/properties/P000174.md

@@ -0,0 +1,13 @@
+---
+space: S000157
+property: P000174
+value: false
+refs:
+- mathse: 342091
+  name: Why does the union of a chain of countable sets have cardinality at most $\aleph_1$?
+---
+
+Suppose by contradiction that $X$ is well-based at the point $p$.
+So there is a chain (totally ordered by inclusion) of neighborhoods of $p$ forming a local base at $p$.
+Their complements form a chain $\mathscr C$ of countable subsets of $X\setminus\{p\}$ whose union is $X\setminus\{p\}$.
+But this is impossible since, as shown for example in {{mathse:342091}}, $|\bigcup\mathscr C|\le\aleph_1<\aleph_2$.

spaces/S000157/properties/P000203.md

@@ -0,0 +1,7 @@
+---
+space: S000157
+property: P000203
+value: true
+---
+
+All points except $p$ are isolated.