Update S132

spaces/S000072/README.md

@@ -6,15 +6,21 @@ refs:
   - doi: 10.1007/978-1-4612-6290-9
     name: Counterexamples in Topology
 ---
-Let $S=((0,1)\setminus\{\frac{1}{2}\})\times(0,1)$ and $X=S\cup\{\langle 0,0\rangle,\langle 1,0\rangle\}\cup\{\langle\frac{1}{2},r\sqrt{2}\rangle:r\in(0,\frac{1}{\sqrt 2})\cap\mathbb Q)\}$.
 
-This space is $X$ where $S$ has its subspace topology from {S176}. Then let
+Let $S=\big((0,1)_{\mathbb Q}\setminus\{\frac{1}{2}\}\big)\times(0,1)_{\mathbb Q}$ and
+$X=S\cup\{\langle 0,0\rangle,\langle 1,0\rangle\}\cup\{\langle\frac{1}{2},r\sqrt{2}\rangle:r\in(0,\frac{1}{\sqrt 2})_{\mathbb Q}\}$
+(where $I_{\mathbb Q}=I\cap\mathbb Q$).
 
-- $U_n(0,0) = \{(0,0)\} \cup \{(x,y): 0<x<\frac{1}{4}, 0<y<\frac{1}{n}\}$,
-- $U_n(1,0) = \{(1,0)\} \cup \{(x,y): \frac{3}{4}<x<1, 0<y<\frac{1}{n}\}$, and
-- $U_n(\frac{1}{2},r\sqrt{2}) = \{(x,y): \frac{1}{4}<x<\frac{3}{4}, |y - r\sqrt{2}|<\frac{1}{n}\}$
+This space is $X$ with the topology such that points of $S$ have their usual neighborhoods as a subspace of {S176}, and
 
-be local bases at $(0,0)$, $(1,0)$ and $(\frac{1}{2},r\sqrt{2})$ respectively.
+- $U_n(0,0) = \{\langle 0,0\rangle\} \cup \left((0,\frac{1}{4})_{\mathbb Q}\times(0,\frac{1}{n})_{\mathbb Q}\right)$,
+- $U_n(1,0) = \{\langle 1,0\rangle\} \cup \left((\frac{3}{4},1)_{\mathbb Q}\times(0,\frac{1}{n})_{\mathbb Q}\right)$, and
+- $U_n(\frac{1}{2},r\sqrt{2}) = \left((\frac{1}{4},\frac{3}{4})\times(r\sqrt{2}-\frac{1}{n},r\sqrt{2}+\frac{1}{n})\right)\cap X$
+  for $r\in(0,\frac{1}{\sqrt 2})_{\mathbb Q}$.
 
-Defined as counterexample #80 ("Arens Square")
-in {{doi:10.1007/978-1-4612-6290-9}}.
+are local bases at $\langle 0,0\rangle$, $\langle 1,0\rangle$ and $\langle\frac{1}{2},r\sqrt{2}\rangle$ respectively.
+
+Defined as counterexample #80 ("Arens Square") in {{doi:10.1007/978-1-4612-6290-9}}, where
+this space was erroneously claimed to be {P4}.
+Compare with {S80} which modifies the construction to
+obtain a valid {P4} example.

spaces/S000129/properties/P000023.md

@@ -7,5 +7,6 @@ refs:
   name: Counterexamples in Topology
 ---
 
-Asserted in the General Reference Chart for space #132 in
-{{doi:10.1007/978-1-4612-6290-9_6}}.
+Consider any neighborhood $U$ of a point $p\in\mathbb S^1\subset X$. For sufficiently small $\epsilon>0$
+the set $A=\{x\in U: \|x\|=1-\epsilon\}$ is infinite (actually of cardinality continuum).
+Moreover, $A$ is closed and discrete by definition of $X$, hence $U\supset A$ cannot be compact.

spaces/S000129/properties/P000043.md

@@ -0,0 +1,10 @@
+---
+space: S000129
+property: P000043
+value: true
+refs:
+  - doi: 10.1007/978-1-4612-6290-9
+    name: Counterexamples in Topology
+---
+
+The balls with respect to the metric provided in item #3 for space #132 in {{doi:10.1007/978-1-4612-6290-9}} are arc connected.

spaces/S000129/properties/P000055.md

@@ -0,0 +1,29 @@
+---
+space: S000129
+property: P000055
+value: true
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+Define $h:X\to[0,+\infty)$ as $h(x)=\frac{1}{\|x\|}-1$ and $r:X\to \mathbb S^1$ as $r(x):=\frac{1}{\|x\|}x$. These are clearly continuous maps on $X$ (the topology is finer than Euclidian). Then consider a metric $d$ given by:
+- $d(p,q)=|h(p)-h(q)|$, if $r(p)=r(q)$;
+- $d(p,q)=h(p)+h(q)+\|r(p)-r(q)\|$, otherwise.
+
+Symmetry and reflexivity of $d$ is immediate. Triangle inequality requires consideration of few cases but is elementary as well.
+
+By continuity of $r$ and $h$ we obtain that convergence in the topology on $X$ implies convergence with respect to $d$.
+
+Assume $d(x_n,x)\to 0$ for some $x\in X$ and $(x_n)\subset X$. If $h(x)>0$, then clearly $r(x_n)=r(x)$ for sufficeintly large $n$ and $\|x_n\|\to \|x\|$,
+implying convergence $x_n \to x$ in $X$.
+For $h(x)=0$, we have $h(x_n)\to 0$ and $r(x_n)\to r(x)=x$ in the Euclidean topology. But since $x$ inherits Euclidean Neihgbourhoods we obtain convergence $x_n\to x$ in this case as well.
+Hence $d$ is indeed a metric providing the topology on $X$.
+
+To prove completeness of $d$, observe that
+a Cauchy sequence $(x_n)\subset X$ with respect to $d$ either satisfies $h(x_n)\to 0$ or is eventually contained in one of the
+open radii. Since any radius (i.e. $(0,1]\cdot p$ with $p\in \mathbb S^1$) is isometric with $[0,+\infty)$, a Cauchy sequence contained therein has a limit.
+Now focus on the case $h(x_n)\to 0$. Clearly,
+also $(r(x_n))$ has to be a Cauchy sequence.
+Since Euclidean metric on $\mathbb S^1$ is complete, there exists $z:=\lim_{n\to \infty} r(x_n)$. Since also $h(x_n)\to 0$, we obtain
+$d(x_n,z)\to 0$, which ends the proof.

spaces/S000129/properties/P000086.md

@@ -0,0 +1,8 @@
+---
+space: S000129
+property: P000086
+value: false
+---
+
+{S129|P23})
+but the point $(0,1/2)$ has a neighbourhood which is a (compact) segment.

spaces/S000129/properties/P000089.md

@@ -0,0 +1,7 @@
+---
+space: S000129
+property: P000089
+value: false
+---
+
+The map $(x,y)\mapsto (-x,-y)$ is a homeomorphism and has no fixed point.

spaces/S000129/properties/P000166.md

@@ -0,0 +1,8 @@
+---
+space: S000129
+property: P000166
+value: true
+---
+
+The topology on $X$ is clearly finer than the Euclidean topology. The set $X$ with
+the Euclidean topology is separable.

spaces/S000129/properties/P000198.md

@@ -4,4 +4,4 @@ property: P000198
 value: false
 ---
 
-The circle $C = \left\{ (x, y) \in \mathbb R^2 \middle| x^2 + y^2 = \frac 12 \right\}$ is a closed discrete subset with cardinality $\mathfrak c$.
+$A=\{x\in X: \|x\|=1/2\}$ is a closed discrete and uncountable subset of $X$.

spaces/S000129/properties/P000200.md

@@ -0,0 +1,8 @@
+---
+space: S000129
+property: P000200
+value: false
+---
+
+The map $X\ni p\mapsto \frac{1}{\|p\|}p \in \mathbb S^1$ is clearly a retraction of $X$ onto the unit circle.
+Hence the fundamental group of $X$ contains a subgroup isomorphic to $\pi_1(\mathbb S^1)\simeq\mathbb Z$ and therefore is nontrivial.

spaces/S000129/properties/P000205.md

@@ -7,4 +7,6 @@ refs:
   name: Counterexamples in Topology
 ---
 
-See item #4 for space #132 in {{doi:10.1007/978-1-4612-6290-9_6}}.
+The space is {P36}; see item #2 for space #132 in {{doi:10.1007/978-1-4612-6290-9_6}}.
+
+And for every $p\in X$ the open segment (radius) $\{\lambda p: 0< \lambda < 1 \}$ is both a closed and open subset of $X\setminus\{p\}$. Hence $p$ is a cut point.

spaces/S000132/properties/P000056.md

@@ -0,0 +1,10 @@
+---
+space: S000132
+property: P000056
+value: true
+refs:
+- doi: 10.2307/2309171
+  name: A Topology for Sequences of Integers II
+---
+
+See Section 3 of {{doi:10.2307/2309171}.

spaces/S000139/README.md

@@ -4,7 +4,7 @@ name: Countable bouquet of circles
 aliases:
 - $\mathbb R$ with $\mathbb Z$ collapsed to a point
 - Countable wedge sum of circles
-- Rose with countably infinite petals
+- Rose with countably many petals
 refs:
   - mathse: 4844916
     name: Answer to "Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?"
@@ -19,4 +19,5 @@ Let $X=\mathbb R/\mathbb Z$ to be the quotient of $\mathbb R$ (with its Euclidea
 Alternatively, this space can be characterized as the *wedge sum*
 (see {{wikipedia:Wedge_sum}}) of countably-many circles,
 also known as a *bouquet of countably many circles*.
+
 Not to be confused with {S201}, which has a coarser topology.

spaces/S000145/properties/P000044.md

@@ -0,0 +1,12 @@
+---
+space: S000145
+property: P000044
+value: true
+---
+
+Let $Y$ be a {P36} subset of $X$ with at least two points.
+If $Y$ is closed in $X$, by the property of ultrafilters all subsets of $Y$ are closed, thus $Y$ is {P52}, which is not possible.
+This implies that $Y$ is open, that is, every {P36} subsets of $X$ must be open.
+
+Now, because $X$ is {P36}, $X$ cannot be partitioned into two nonempty open subsets.
+We conclude that $X$ cannot be partitioned into two {P36} subsets, each with at least two points.

spaces/S000145/properties/P000086.md

@@ -0,0 +1,10 @@
+---
+space: S000145
+property: P000086
+value: true
+---
+
+Let $p, q \in X$ be arbitrary distinct points.
+Define $f(x) = \begin{cases} q & \text{if }x = p, \\ p & \text{if }x = q, \\ x & \text{otherwise}. \end{cases}$
+
+Then $f(x)$ is involutary and continuous since for every open subset $U$ and every point $x \in X$, both $U \setminus \{x\} = U \cap \left( X \setminus \{x\} \right)$ and $U \cup \{x\}$ are open.

spaces/S000145/properties/P000089.md

@@ -0,0 +1,10 @@
+---
+space: S000145
+property: P000089
+value: true
+refs:
+  - mathse: 5021484
+    name: Does a free ultrafilter topology have the fixed point property?
+---
+
+See {{mathse:5021484}}.

spaces/S000145/properties/P000101.md

@@ -0,0 +1,12 @@
+---
+space: S000145
+property: P000101
+value: false
+---
+
+Let $p, q \in X$ be arbitrary distinct points.
+Define $f(x) = \begin{cases} q & \text{if }x = p, \\ x & \text{otherwise}. \end{cases}$
+
+Then $f(x)$ is continuous since for every open subset $U$ and every point $x \in X$, both $U \setminus \{x\} = U \cap \left( X \setminus \{x\} \right)$ and $U \cup \{x\}$ are open.
+
+Therefore, $f$ is a retract from $X$ to $X \setminus \{p\}$, which is not closed.

spaces/S000145/properties/P000138.md

@@ -0,0 +1,12 @@
+---
+space: S000145
+property: P000138
+value: false
+---
+
+Let $A$ be an infinite closed proper subset of $X$, and take an arbitrary $p \in X \setminus A$.
+For each $S \subseteq A$, the function $f_S:X\to X$ defined by
+$$f_S(x) = \begin{cases} p & \text{if }x\in S, \\ x & \text{otherwise}, \end{cases}$$
+is continuous since for every nonempty open set $U$, its inverse image $f^{-1}(U)$ contains $U\setminus A$, which is open and nonempty; hence $f^{-1}(U)$ is open.
+
+And there are continuum many such functions $f_S$.

spaces/S000201/README.md

@@ -9,6 +9,10 @@ refs:
   name: Hawaiian earring on Wikipedia
 - zb: "0951.54001"
   name: Topology (Munkres)
+- zb: "1044.55001"
+  name: Algebraic Topology (Hatcher)
+- doi: 10.1007/978-1-4612-4576-6
+  name: An Introduction to Algebraic Topology (Rotman)
 ---
 
 The subspace of {S176} defined by
@@ -20,5 +24,5 @@ integers $n$.
 
 Not to be confused with {S139}, which has a finer topology.
 
-Defined as "infinite earring" in section 71 of {{zb:0951.54001}}.
-Appears as Example 1.25 in {{zb:1044.55001}} (available [here](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)) under the name "Shrinking wedge of circles"; however, note that it is not actually an example of a "wedge sum" as defined in e.g. {{wikipedia:Wedge_sum}}, {{doi:10.1007/978-1-4612-4576-6}}.
+Defined as "infinite earring" in section 71 of {{zb:0951.54001}}; usually known in the literature as "Hawaiian earring".
+Appears as Example 1.25 in {{zb:1044.55001}} (available [here](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)) under the name "Shrinking wedge of circles"; however, note that it is not actually an example of a "wedge sum" as defined in e.g. {{wikipedia:Wedge_sum}} or {{doi:10.1007/978-1-4612-4576-6}}.