update master

CITATION.cff

@@ -9,7 +9,6 @@ message: >-
 type: dataset
 authors:
   - given-names: The pi-Base Community
-
 repository-code: 'https://github.com/pi-base/data'
 url: 'https://topology.pi-base.org/'
 abstract: A community database of topological counterexamples

properties/P000001.md

@@ -4,6 +4,7 @@ slug: t_0
 name: "$T_0$"
 aliases:
   - Kolmogorov
+  - T0
 refs:
   - doi: 10.1007/978-1-4612-6290-9
     name: Counterexamples in Topology

properties/P000002.md

@@ -4,6 +4,7 @@ slug: t_1
 name: "$T_1$"
 aliases:
   - Fréchet
+  - T1
 refs:
   - doi: 10.1007/978-1-4612-6290-9
     name: Counterexamples in Topology

properties/P000003.md

@@ -4,6 +4,7 @@ slug: t_2
 name: "$T_2$"
 aliases:
   - Hausdorff
+  - T2
 refs:
   - doi: 10.1007/978-1-4612-6290-9
     name: Counterexamples in Topology

properties/P000004.md

@@ -5,6 +5,7 @@ name: "$T_{2 \\frac{1}{2}}$"
 aliases:
   - Urysohn
   - Completely Hausdorff
+  - T2.5
 refs:
   - mr: MR2048350
     name: General Topology (Willard)

properties/P000005.md

@@ -4,6 +4,7 @@ slug: t_3
 name: "$T_3$"
 aliases:
   - Regular Hausdorff
+  - T3
 refs:
   - mr: MR2048350
     name: General Topology (Willard)

properties/P000006.md

@@ -5,6 +5,7 @@ name: "$T_{3 \\frac{1}{2}}$"
 aliases:
   - Tychonoff
   - Completely Regular Hausdorff
+  - T3.5
 refs:
   - mr: MR2048350
     name: General Topology (Willard)

properties/P000007.md

@@ -4,6 +4,7 @@ slug: t_4
 name: "$T_4$"
 aliases:
   - Normal Hausdorff
+  - T4
 refs:
   - mr: MR2048350
     name: General Topology (Willard)

properties/P000008.md

@@ -4,6 +4,7 @@ slug: t_5
 name: "$T_5$"
 aliases:
   - Completely normal Hausdorff
+  - T5
 refs:
   - doi: 10.1007/978-1-4612-6290-9
     name: Counterexamples in Topology

properties/P000035.md

@@ -2,6 +2,8 @@
 uid: P000035
 slug: fully-t_4
 name: Fully $T_4$
+aliases:
+  - Fully T4
 refs:
   - doi: 10.1007/978-1-4612-6290-9
     name: Counterexamples in Topology

properties/P000043.md

@@ -6,6 +6,6 @@ refs:
   - doi: 10.1007/978-1-4612-6290-9
     name: Counterexamples in Topology
 ---
-A space $X$ is locally connected if $X$ has a basis consisting of arc connected sets.
+A space with a basis consisting of (open) {P38} sets.  Equivalently, every point has a local base of {P38} open neighborhoods.
 
 Defined on page 30 of {{doi:10.1007/978-1-4612-6290-9}}.

properties/P000067.md

@@ -5,6 +5,7 @@ name: "$T_6$"
 aliases:
   - Perfectly T_4
   - Perfectly normal Hausdorff
+  - T6
 refs:
   - doi: 10.1007/978-1-4612-6290-9
     name: Counterexamples in Topology

properties/P000086.md

@@ -3,7 +3,7 @@ uid: P000086
 slug: homogenous
 name: Homogeneous
 refs:
-  - mr: 2048350
+  - mr: MR2048350
     name: General Topology (Willard)
 ---
 A space $X$ such that for each $a,b\in X$ there is a homeomorphism $\phi : X \to X$ such that $\phi(a)=b$. In other words: the group of homeomorphisms of the space onto itself acts transitively.

properties/P000098.md

@@ -5,7 +5,7 @@ name: Generated by countably-many compacts
 aliases:
   - $k_\omega$
 refs:
-  - mr: 0540599
+  - mr: MR0540599
     name: A survey of kω-spaces
 ---
 

properties/P000144.md

@@ -1,10 +1,10 @@
----
-uid: P000144
-slug: locally-pseudometrizable
-name: Locally pseudometrizable
-refs:
-  - doi: 10.1090/S0002-9947-1969-0251719-4
-    name: TODO
----
-
-Every point has a neighborhood that is {P121}.
+---
+uid: P000144
+slug: locally-pseudometrizable
+name: Locally pseudometrizable
+refs:
+  - mathse: 4659902
+    name: How can the implication metrizable -> first countable be generalized?
+---
+
+Every point has a neighborhood that is {P121}.

properties/P000145.md

@@ -0,0 +1,12 @@
+---
+uid: P000145
+slug: strongly-paracompact
+name: Strongly paracompact
+refs:
+  - mr: MR1039321
+    name: General Topology (Engelking, 1989)
+---
+
+Every open cover has a star-finite open refinement.
+
+Defined on p. 326 of {{mr:MR1039321}} together with the condition that the space is {P3}.  Here we do not assume any separation axiom as part of the definition.

properties/P000146.md

@@ -0,0 +1,14 @@
+---
+uid: P000146
+slug: ultraparacompact
+name: Ultraparacompact
+refs:
+  - doi: 10.48550/arXiv.1306.6086
+    name: Ultraparacompactness and Ultranormality (J. Van Name)
+  - mr: MR261565
+    name: Extending continuous functions on zero-dimensional spaces (R. Ellis)
+---
+
+Every open cover of the space has an open (hence clopen) refinement partitioning the space. (We do not assume {P3}.)
+
+Equivalently, every open cover of the space is refined by a locally finite clopen cover.  The equivalence between the two definitions is shown in Lemma 1.3 and Corollary 1.4 of {{mr:MR261565}} (<https://eudml.org/doc/161976>).

spaces/S000020/README.md

@@ -9,19 +9,12 @@ refs:
   - doi: 10.1007/978-1-4612-6290-9 
     name: Counterexamples in Topology
   - wikipedia: Fort_space
-    name: Fort space
+    name: Fort space on Wikipedia
 ---
-Let \(X=\omega\cup\{\infty\}=\{0,1,2\dots\}\cup\{\infty\}\).
-Define $U \subset X$ to open if its complement is finite or includes \(\infty\).
+Let $X=\omega\cup\{\infty\}=\{0,1,2\dots\}\cup\{\infty\}$.
+Define $U \subseteq X$ to be open if its complement either is finite or includes $\infty$.
+
+This space is the one-point compactification of a countably infinite discrete space.
 
 Defined as counterexample #23 ("Countable Fort Space")
 in {{doi:10.1007/978-1-4612-6290-9}}.
-
-<!-- [[Proof of Topology]]
-Let $\tau$ be the collection of all open sets $U \subset X$.
-
-Then $X$ is an element of $\tau$ since $X \setminus X = \emptyset$. Also, the empty set is an element of $\tau$ since $p \notin \emptyset$.
-
-Now, let $\mathcal{A}$ be a subcollection of elements from $\tau$, two cases arise. The first case is that no $A \in \mathcal{A}$ contains $p$. Thus, $\bigcup\limits_{A \in \mathcal{A}} A \in \tau.$ The second case is that $p$ is in at least one element of $\mathcal{A}$. Without loss of generality, let $p \in A_1$. Denote $U = \cup_{A \in \mathcal{A}} A$. Now, $X \setminus U \subset X \setminus A_1.$ Since, $p \in A_1$ but $A_1$ is still open, it must be that $X \setminus A_1$ is finite. Therefore, $X \setminus U$ is finite, so $U \in \tau$.
-
-Finally, let $\mathcal{A}$ be a subcollection, two cases arise. In the first case some $A \in \mathcal{A}$ does not have $p$ in it. Then, $p \notin \bigcap\limits_{A \in \mathcal{A}} A$. Thus $p \notin U$, $\in \tau$. In the second case $p \in A$ for all $A \in \mathcal{A}$. We know that $X \setminus A$ is finite for all $A \in \mathcal{A}$. So, $p \in (\,\bigcap\limits_{A \in \mathcal{A}} A$)\, Now consider (\,$\bigcap\limits_{A \in \mathcal{A}} A)\,^c = \bigcup\limits_{A \in \mathcal{A}} A^c$, by DeMorgan's. Thus, it is countable, because a union of a countable set is countable. Showing that the third axiom of a topological space is met. -->

spaces/S000023/README.md

@@ -16,3 +16,5 @@ a finite number of points in all but a finite number of columns.
 
 Defined as counterexample #26 ("Arens-Fort Space")
 in {{doi:10.1007/978-1-4612-6290-9}}.
+
+This space is homeomorphic to a subspace of {S156}.

spaces/S000025/properties/P000068.md

@@ -0,0 +1,12 @@
+---
+space: S000025
+property: P000068
+value: false
+refs:
+- mathse: 740634
+  name: Answer to How to show that R is not Rothberger, and how to show that it is not Menger?
+---
+
+See {{mathse:740634}}: let $\mathcal U_n$ cover $\mathbb R$ with sets of measure $2^{-n}$;
+any single selection from each cover creates a collection covering at most a subset of
+$\mathbb R$ of measure $2$.

spaces/S000025/properties/P000123.md

@@ -3,7 +3,7 @@ space: S000025
 property: P000123
 value: true
 refs:
-- mr: 3728284
+- mr: MR3728284
   name: Topology (Munkres)
 ---
 

spaces/S000031/properties/P000020.md

@@ -3,7 +3,7 @@ space: S000031
 property: P000020
 value: true
 refs:
-  - mr: 2048350
+  - mr: MR2048350
     name: General Topology (Willard)
   - mathse: 1288007
     name: Countable Product of Sequentially Compact spaces is Sequentially Compact

spaces/S000031/properties/P000036.md

@@ -3,7 +3,7 @@ space: S000031
 property: P000036
 value: true
 refs:
-- mr: 2048350
+- mr: MR2048350
   name: General Topology (Willard)
 ---
 

spaces/S000031/properties/P000041.md

@@ -3,7 +3,7 @@ space: S000031
 property: P000041
 value: false
 refs:
-  - mr: 2048350
+  - mr: MR2048350
     name: General Topology (Willard)
 ---
 

spaces/S000031/properties/P000130.md

@@ -3,7 +3,7 @@ space: S000031
 property: P000130
 value: false
 refs:
-  - mr: 2048350
+  - mr: MR2048350
     name: General Topology (Willard)
 ---
 

spaces/S000042/properties/P000068.md

@@ -0,0 +1,10 @@
+---
+space: S000042
+property: P000068
+value: true
+refs:
+- mathse: 3200061
+  name: Answer to Rothberger game and Meager set.
+---
+
+See {{mathse:3200061}}: contain $-n$ from the $n$th cover.

spaces/S000063/properties/P000008.md

@@ -0,0 +1,11 @@
+---
+space: S000063
+property: P000008
+value: true
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+See items #1 and #2 for space #71 in
+{{doi:10.1007/978-1-4612-6290-9_6}}.

spaces/S000063/properties/P000015.md

@@ -0,0 +1,11 @@
+---
+space: S000063
+property: P000015
+value: false
+refs:
+- mathse: 4053603
+  name: Answer to Can the closure of a set be written as the intersection of open neighborhoods in a non-metrizable space?
+---
+
+See {{mathse:4053603}} and
+<https://dantopology.wordpress.com/2012/10/28/michael-line-basics/>.

spaces/S000063/properties/P000030.md

@@ -0,0 +1,11 @@
+---
+space: S000063
+property: P000030
+value: true
+refs:
+- mathse: 3702566
+  name: Answer to Showing R with the standard topology union irrational subsets is normal.
+---
+
+See {{mathse:3702566}} and
+<https://dantopology.wordpress.com/2012/10/28/michael-line-basics/>.

spaces/S000077/README.md

@@ -7,7 +7,8 @@ refs:
   - doi: 10.1007/978-1-4612-6290-9 
     name: Counterexamples in Topology
 ---
-Let $P$ be the set of irrationals in $\mathbb{R}$. Michael's Product topology is the product space $X = \mathbb{R} \times P$ where $\mathbb{R}$ has the discrete irrational extension topology (each irrational is isolated) and $P$ has the subspace topology.
+
+The product of {S63} and {S28}.
 
 Defined as counterexample #85 ("Michael's Product Topology")
 in {{doi:10.1007/978-1-4612-6290-9}}.

spaces/S000077/properties/P000011.md → spaces/S000077/properties/P000006.md

@@ -1,11 +1,13 @@
 ---
 space: S000077
-property: P000011
+property: P000006
 value: true
 refs:
 - doi: 10.1007/978-1-4612-6290-9_6
   name: Counterexamples in Topology
 ---
 
+Follows as this space is the product of {P000006} spaces.
+
 Asserted in the General Reference Chart for space #85 in
 {{doi:10.1007/978-1-4612-6290-9_6}}.

spaces/S000125/README.md

@@ -1,17 +1,17 @@
 ---
 uid: S000125
-slug: cantor's-leaky-tent
-name: Cantor's leaky tent
+slug: kn-fan
+name: Knaster-Kuratowski fan
 aliases:
-  - Knaster-Kuratowski fan
+  - Cantor's leaky tent
 counterexamples_id: 128
 refs:
 - doi: 10.1007/978-1-4612-6290-9
   name: Counterexamples in Topology
 - wikipedia: Knaster-Kuratowski_fan
   name: Knaster-Kuratowski fan
 ---
-For any $a \in [0,1]$, let $L(a)$ be the line segment from $(a,0)$ to $p = (\frac{1}{2}, \frac{1}{2})$. Let $\mathcal{C}$ be the middle-thirds Cantor set in the unit interval, $\mathcal{E}$ the endpoints of the removed intervals and $\mathcal{F} = \mathcal{C} \setminus \mathcal{E}$. Define $A = \{(x,y) \in L(c)\ |\ c \in \mathcal{E}, y \in \mathbb{Q}\}$ and $B = \{(x,y) \in L(c)\ |\ c \in \mathcal{F}, y \not\in \mathbb{Q}\}$. Cantor's Leaky Tent is $X = A \cup B \subset \mathbb{R}^2$ with the subspace topology.
+For any $a \in [0,1]$, let $L(a)$ be the line segment from $(a,0)$ to $p = (\frac{1}{2}, \frac{1}{2})$. Let $\mathcal{C}$ be the middle-thirds Cantor set in the unit interval, $\mathcal{E}$ the endpoints of the removed intervals and $\mathcal{F} = \mathcal{C} \setminus \mathcal{E}$. Define $A = \{(x,y) \in L(c)\ |\ c \in \mathcal{E}, y \in \mathbb{Q}\}$ and $B = \{(x,y) \in L(c)\ |\ c \in \mathcal{F}, y \not\in \mathbb{Q}\}$. This space is $X = A \cup B \subset \mathbb{R}^2$ with the subspace topology.
 
 Defined as counterexample #128 ("Cantor's Leaky Tent")
 in {{doi:10.1007/978-1-4612-6290-9}}.

spaces/S000126/README.md

@@ -1,17 +1,17 @@
 ---
 uid: S000126
-slug: cantor's-teepee
-name: Cantor's Teepee
+slug: punctured-kn-fan
+name: Punctured Knaster-Kuratowski fan
 aliases:
-  - Punctured Knaster-Kuratowski fan
+  - Cantor's Teepee
 counterexamples_id: 129
 refs:
 - doi: 10.1007/978-1-4612-6290-9
   name: Counterexamples in Topology
 - wikipedia: Knaster-Kuratowski_fan
   name: Knaster-Kuratowski fan
 ---
-Cantor's Teepee is Cantor's Leaky Tent with the apex removed. Specifically: For any $a \in [0,1]$, let $L(a)$ be the line segment from $(a,0)$ to $p = (\frac{1}{2}, \frac{1}{2})$. Let $\mathcal{C}$ be the middle-thirds Cantor set in the unit interval, $\mathcal{E}$ the endpoints of the removed intervals and $\mathcal{F} = \mathcal{C} \setminus \mathcal{E}$. Define $A = \{(x,y) \in L(c)\ |\ c \in \mathcal{E}, y \in \mathbb{Q}\}$ and $B = \{(x,y) \in L(c)\ |\ c \in \mathcal{F}, y \not\in \mathbb{Q}\}$. Cantor's Teepee is $X = A \cup B \setminus\{p\} \subset \mathbb{R}^2$ with the subspace topology.
+{S000125} with the apex removed. Specifically: For any $a \in [0,1]$, let $L(a)$ be the line segment from $(a,0)$ to $p = (\frac{1}{2}, \frac{1}{2})$. Let $\mathcal{C}$ be the middle-thirds Cantor set in the unit interval, $\mathcal{E}$ the endpoints of the removed intervals and $\mathcal{F} = \mathcal{C} \setminus \mathcal{E}$. Define $A = \{(x,y) \in L(c)\ |\ c \in \mathcal{E}, y \in \mathbb{Q}\}$ and $B = \{(x,y) \in L(c)\ |\ c \in \mathcal{F}, y \not\in \mathbb{Q}\}$. This space is $X = (A \cup B) \setminus\{p\} \subset \mathbb{R}^2$ with the subspace topology.
 
 Defined as counterexample #129 ("Cantor's Teepee")
 in {{doi:10.1007/978-1-4612-6290-9}}.

spaces/S000153/README.md

@@ -0,0 +1,16 @@
+---
+uid: S000153
+slug: open-long-ray
+name: Open long ray
+aliases:
+  - Open long line
+refs:
+  - wikipedia: Long_line_(topology)
+    name: Long line on Wikipedia
+  - mr: MR0464128
+    name: Topology (Munkres)
+---
+
+The space obtained by taking $Y=\omega_1 \times [0,1)$ with its lexicographic order topology ({S38}) and removing the first element $(0,0)$.
+
+The resulting space $X$ is an open subspace of $Y$ with the subspace topology.  This is the same as the order topology on $X$, since $X$ is order-convex in $Y$ (see Theorem 16.4 in {{mr:0464128}}).