From 2d59ca2b625fd698a4b5dd81174cf7a1da480ffd Mon Sep 17 00:00:00 2001 From: Batixx Date: Thu, 26 Mar 2026 17:49:47 +0100 Subject: [PATCH 1/9] s171 p22 --- spaces/S000171/properties/P000022.md | 7 +++++++ 1 file changed, 7 insertions(+) create mode 100644 spaces/S000171/properties/P000022.md diff --git a/spaces/S000171/properties/P000022.md b/spaces/S000171/properties/P000022.md new file mode 100644 index 0000000000..2fac9905c8 --- /dev/null +++ b/spaces/S000171/properties/P000022.md @@ -0,0 +1,7 @@ +--- +space: S000171 +property: P000022 +value: false +--- + +This topology refines {S25}, therefore the identity to $\mathbb{R}$ is continuous and unbounded. From 9adb26e56820817ed6d845808c5a4fbd55b3848b Mon Sep 17 00:00:00 2001 From: Batixx Date: Mon, 30 Mar 2026 21:03:26 +0200 Subject: [PATCH 2/9] add def --- spaces/S000171/README.md | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) diff --git a/spaces/S000171/README.md b/spaces/S000171/README.md index eef348b0c4..3764f2da7c 100644 --- a/spaces/S000171/README.md +++ b/spaces/S000171/README.md @@ -4,7 +4,14 @@ name: Brian's Example refs: - mo: 416331 name: Example of an uncountable scattered space with some properties + - wikipedia: Bernstein_set + - name: Bernstein set on Wikipedia --- -In {{mo:416331}} Will Brian provides an example in ZFC for an uncountable, Hausdorff, +Let $X = \mathbb{R}$ and let $(X_i)_{i\in \omega}$ be a partition of $X$ into [Bernstein sets](https://en.wikipedia.org/wiki/Bernstein_set) (for the Euclidean topology). +Given some $x \in X$, there is a unique $n \in \omega$ such that $x \in X_n$. Then $\{x\} \cup (U \cap \bigcup_{i Date: Mon, 30 Mar 2026 21:03:42 +0200 Subject: [PATCH 3/9] typo --- spaces/S000171/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000171/README.md b/spaces/S000171/README.md index 3764f2da7c..b0d4c61cf1 100644 --- a/spaces/S000171/README.md +++ b/spaces/S000171/README.md @@ -9,7 +9,7 @@ refs: --- Let $X = \mathbb{R}$ and let $(X_i)_{i\in \omega}$ be a partition of $X$ into [Bernstein sets](https://en.wikipedia.org/wiki/Bernstein_set) (for the Euclidean topology). -Given some $x \in X$, there is a unique $n \in \omega$ such that $x \in X_n$. Then $\{x\} \cup (U \cap \bigcup_{i Date: Mon, 30 Mar 2026 21:05:07 +0200 Subject: [PATCH 4/9] redundant - --- spaces/S000171/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000171/README.md b/spaces/S000171/README.md index b0d4c61cf1..7e77d75a84 100644 --- a/spaces/S000171/README.md +++ b/spaces/S000171/README.md @@ -5,7 +5,7 @@ refs: - mo: 416331 name: Example of an uncountable scattered space with some properties - wikipedia: Bernstein_set - - name: Bernstein set on Wikipedia + name: Bernstein set on Wikipedia --- Let $X = \mathbb{R}$ and let $(X_i)_{i\in \omega}$ be a partition of $X$ into [Bernstein sets](https://en.wikipedia.org/wiki/Bernstein_set) (for the Euclidean topology). From f9e932abc04307a44caf96af2a2cb3a1ab400f82 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Fri, 15 May 2026 17:34:14 +0200 Subject: [PATCH 5/9] rename space --- spaces/S000171/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000171/README.md b/spaces/S000171/README.md index 7e77d75a84..5fcead035c 100644 --- a/spaces/S000171/README.md +++ b/spaces/S000171/README.md @@ -1,6 +1,6 @@ --- uid: S000171 -name: Brian's Example +name: Brian's stack of Bernstein sets refs: - mo: 416331 name: Example of an uncountable scattered space with some properties From d8f4c932b02c36988d1761d47ce349d54a23c2fa Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Fri, 3 Jul 2026 03:48:22 +0200 Subject: [PATCH 6/9] Update spaces/S000171/properties/P000022.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- spaces/S000171/properties/P000022.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000171/properties/P000022.md b/spaces/S000171/properties/P000022.md index 2fac9905c8..08a77c2524 100644 --- a/spaces/S000171/properties/P000022.md +++ b/spaces/S000171/properties/P000022.md @@ -4,4 +4,4 @@ property: P000022 value: false --- -This topology refines {S25}, therefore the identity to $\mathbb{R}$ is continuous and unbounded. +{S25} has a coarser topology than $X$, but {S25|P22}. From 0b3bb151b96de676b5d0c7d1e9fb810639eb9ba1 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Fri, 3 Jul 2026 07:15:04 +0200 Subject: [PATCH 7/9] Update spaces/S000171/README.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- spaces/S000171/README.md | 9 ++++++--- 1 file changed, 6 insertions(+), 3 deletions(-) diff --git a/spaces/S000171/README.md b/spaces/S000171/README.md index 5fcead035c..a37c682a4a 100644 --- a/spaces/S000171/README.md +++ b/spaces/S000171/README.md @@ -8,10 +8,13 @@ refs: name: Bernstein set on Wikipedia --- -Let $X = \mathbb{R}$ and let $(X_i)_{i\in \omega}$ be a partition of $X$ into [Bernstein sets](https://en.wikipedia.org/wiki/Bernstein_set) (for the Euclidean topology). -Given some $x \in X$, there is a unique $n \in \omega$ such that $x \in X_n$. Then $\{x\} \cup (U \cap \bigcup_{i Date: Fri, 3 Jul 2026 12:49:01 +0200 Subject: [PATCH 8/9] link to answer --- spaces/S000171/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000171/README.md b/spaces/S000171/README.md index a37c682a4a..f4701a6bfa 100644 --- a/spaces/S000171/README.md +++ b/spaces/S000171/README.md @@ -16,5 +16,5 @@ with $U$ (basic) open neighbourhood of $x$ in the Euclidean topology. This topology is finer than {S25}. -This space was constructed in {{mo:416331}} by Will Brian, providing an example in ZFC for an uncountable, Hausdorff, +This space was constructed in {{mo:416752}} by Will Brian, providing an example in ZFC for an uncountable, Hausdorff, first-countable, scattered, Lindelöf space. From b7fb6d75d99a497147f3782fa51e25b860792c60 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Fri, 3 Jul 2026 16:15:34 -0400 Subject: [PATCH 9/9] update refs --- spaces/S000171/README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/spaces/S000171/README.md b/spaces/S000171/README.md index f4701a6bfa..876a2f80e5 100644 --- a/spaces/S000171/README.md +++ b/spaces/S000171/README.md @@ -2,8 +2,8 @@ uid: S000171 name: Brian's stack of Bernstein sets refs: - - mo: 416331 - name: Example of an uncountable scattered space with some properties + - mo: 416752 + name: Answer to "Example of an uncountable scattered space with some properties" - wikipedia: Bernstein_set name: Bernstein set on Wikipedia ---