From 484a4aa0e9e4a0c7330988bc383076edfb1207a2 Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Sat, 18 Jan 2025 22:54:11 +0100 Subject: [PATCH 01/14] introduction of the space and basic properties --- spaces/S000208/README.md | 15 +++++++++++++++ spaces/S000208/properties/P000006.md | 7 +++++++ spaces/S000208/properties/P000007.md | 7 +++++++ spaces/S000208/properties/P000049.md | 10 ++++++++++ spaces/S000208/properties/P000162.md | 10 ++++++++++ 5 files changed, 49 insertions(+) create mode 100644 spaces/S000208/README.md create mode 100644 spaces/S000208/properties/P000006.md create mode 100644 spaces/S000208/properties/P000007.md create mode 100644 spaces/S000208/properties/P000049.md create mode 100644 spaces/S000208/properties/P000162.md diff --git a/spaces/S000208/README.md b/spaces/S000208/README.md new file mode 100644 index 0000000000..439102fc37 --- /dev/null +++ b/spaces/S000208/README.md @@ -0,0 +1,15 @@ +--- +uid: S000208 +name: Katětov's non-normal subspace of $\beta\mathbb{N}$ +refs: + - doi: 10.1007/978-1-4615-7819-2 + name: Rings of Continuous Functions (Gillman & Jerison) +--- + +Constructed in exercise 6Q of {{doi:10.1007/978-1-4615-7819-2}}. Fix a bijection $\varphi:\mathbb{N}\to\mathbb{Q}$. For each irrational $r$ fix a sequence of rational numbers $s_n\to r$, and let $E_r = \{\varphi^{-1}(s_n) : n\in\mathbb{N}\}$. Let $\mathcal{E} = \{E_r : r\in\mathbb{R}\setminus\mathbb{Q}\}$. Then $\mathcal{E}$ is an almost disjoint family on $\mathbb{N}$ of size $\mathfrak{c}$. + +Let $E'$ be the set of limit points of $E$ in {S108}. Then $E'\neq \emptyset$ and $E_1'\cap E_2' = \emptyset$ for $E, E_1, E_2\in\mathcal{E}$. For each $E\in\mathcal{E}$ take $p_E\in E'$ and let $\Pi = \mathbb{N}\cup D$ where $D = \{p_E : E\in\mathcal{E}\}$. + +Katětov's non-normal subspace of $\beta\mathbb{N}$ is the space $\Pi$. + + diff --git a/spaces/S000208/properties/P000006.md b/spaces/S000208/properties/P000006.md new file mode 100644 index 0000000000..c2d3f27cba --- /dev/null +++ b/spaces/S000208/properties/P000006.md @@ -0,0 +1,7 @@ +--- +space: S000208 +property: P000006 +value: true +--- + +Is a subspace of {S108} and {S108|P6}. diff --git a/spaces/S000208/properties/P000007.md b/spaces/S000208/properties/P000007.md new file mode 100644 index 0000000000..7c92b5a026 --- /dev/null +++ b/spaces/S000208/properties/P000007.md @@ -0,0 +1,7 @@ +--- +space: S000208 +property: P000007 +value: false +--- + +$D$ is a closed discrete subspace of $\Pi$ of size $\mathfrak{c}$. Since there is $2^\mathfrak{c}$ continuous real-valued functions on $D$ and at most $\mathfrak{c}$ continuous real-valued functions on $\Pi$, from Tietze extension theorem $\Pi$ cannot be $T_4$. \ No newline at end of file diff --git a/spaces/S000208/properties/P000049.md b/spaces/S000208/properties/P000049.md new file mode 100644 index 0000000000..2a2a1a8834 --- /dev/null +++ b/spaces/S000208/properties/P000049.md @@ -0,0 +1,10 @@ +--- +space: S000208 +property: P000049 +value: true +refs: +- zb: "0684.54001" + name: General Topology (Engelking, 1989) +--- + +It's a dense subspace of {S108} and {S108|P49}. A dense subspace of extremally disconnected space is extremally disconnected (see {{zb:"0684.54001"}} exercise 6.2.G.c). \ No newline at end of file diff --git a/spaces/S000208/properties/P000162.md b/spaces/S000208/properties/P000162.md new file mode 100644 index 0000000000..d349279bd6 --- /dev/null +++ b/spaces/S000208/properties/P000162.md @@ -0,0 +1,10 @@ +--- +space: S000208 +property: P000162 +value: true +refs: + - doi: 10.1007/978-1-4615-7819-2 + name: Rings of Continuous Functions (Gillman & Jerison) +--- + +Extend $\varphi$ to $\Pi$ so that if $E = E_r$ and $s_n\to r$ then $\varphi(p_E) = \lim_{n\to\infty} \varphi(s_n)$. If $\varphi(p_E)\in U$ where $U\subseteq \mathbb{R}$ is open, find $N$ such that $s_n\in U$ for $n\geq N$, then $V =\overline{E}\setminus\varphi^{-1}(\{s_1, s_2, ..., s_N\})$ is an open neighbourhood of $p_E$ and $\varphi(V)\subseteq U$. This shows that $\varphi:\Pi\to\mathbb{R}$ is continuous. By definition it's clearly a bijection. From corollary 8.18 of {{10.1007/978-1-4615-7819-2}}, $\Pi$ is realcompact. \ No newline at end of file From 50e3c334776a32c9d213b12b9ebc9e782fff2b6b Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Fri, 29 Aug 2025 21:02:30 +0200 Subject: [PATCH 02/14] updated space id --- spaces/{S000208 => S000214}/README.md | 2 +- spaces/{S000208 => S000214}/properties/P000006.md | 2 +- spaces/{S000208 => S000214}/properties/P000007.md | 2 +- spaces/{S000208 => S000214}/properties/P000049.md | 2 +- spaces/{S000208 => S000214}/properties/P000162.md | 2 +- 5 files changed, 5 insertions(+), 5 deletions(-) rename spaces/{S000208 => S000214}/README.md (98%) rename spaces/{S000208 => S000214}/properties/P000006.md (83%) rename spaces/{S000208 => S000214}/properties/P000007.md (95%) rename spaces/{S000208 => S000214}/properties/P000049.md (94%) rename spaces/{S000208 => S000214}/properties/P000162.md (97%) diff --git a/spaces/S000208/README.md b/spaces/S000214/README.md similarity index 98% rename from spaces/S000208/README.md rename to spaces/S000214/README.md index 439102fc37..4c29852258 100644 --- a/spaces/S000208/README.md +++ b/spaces/S000214/README.md @@ -1,5 +1,5 @@ --- -uid: S000208 +uid: S000214 name: Katětov's non-normal subspace of $\beta\mathbb{N}$ refs: - doi: 10.1007/978-1-4615-7819-2 diff --git a/spaces/S000208/properties/P000006.md b/spaces/S000214/properties/P000006.md similarity index 83% rename from spaces/S000208/properties/P000006.md rename to spaces/S000214/properties/P000006.md index c2d3f27cba..c241b2e366 100644 --- a/spaces/S000208/properties/P000006.md +++ b/spaces/S000214/properties/P000006.md @@ -1,5 +1,5 @@ --- -space: S000208 +space: S000214 property: P000006 value: true --- diff --git a/spaces/S000208/properties/P000007.md b/spaces/S000214/properties/P000007.md similarity index 95% rename from spaces/S000208/properties/P000007.md rename to spaces/S000214/properties/P000007.md index 7c92b5a026..9ccc85834a 100644 --- a/spaces/S000208/properties/P000007.md +++ b/spaces/S000214/properties/P000007.md @@ -1,5 +1,5 @@ --- -space: S000208 +space: S000214 property: P000007 value: false --- diff --git a/spaces/S000208/properties/P000049.md b/spaces/S000214/properties/P000049.md similarity index 94% rename from spaces/S000208/properties/P000049.md rename to spaces/S000214/properties/P000049.md index 2a2a1a8834..2bad15a8b0 100644 --- a/spaces/S000208/properties/P000049.md +++ b/spaces/S000214/properties/P000049.md @@ -1,5 +1,5 @@ --- -space: S000208 +space: S000214 property: P000049 value: true refs: diff --git a/spaces/S000208/properties/P000162.md b/spaces/S000214/properties/P000162.md similarity index 97% rename from spaces/S000208/properties/P000162.md rename to spaces/S000214/properties/P000162.md index d349279bd6..766ce02181 100644 --- a/spaces/S000208/properties/P000162.md +++ b/spaces/S000214/properties/P000162.md @@ -1,5 +1,5 @@ --- -space: S000208 +space: S000214 property: P000162 value: true refs: From de1119b850ec76c84e2920aafe200f0c2925726f Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Tue, 16 Dec 2025 23:17:06 +0100 Subject: [PATCH 03/14] changed from S214 to S216 --- spaces/{S000214 => S000216}/README.md | 4 +--- spaces/{S000214 => S000216}/properties/P000006.md | 2 +- spaces/{S000214 => S000216}/properties/P000007.md | 2 +- spaces/{S000214 => S000216}/properties/P000049.md | 2 +- spaces/{S000214 => S000216}/properties/P000162.md | 2 +- 5 files changed, 5 insertions(+), 7 deletions(-) rename spaces/{S000214 => S000216}/README.md (98%) rename spaces/{S000214 => S000216}/properties/P000006.md (83%) rename spaces/{S000214 => S000216}/properties/P000007.md (95%) rename spaces/{S000214 => S000216}/properties/P000049.md (94%) rename spaces/{S000214 => S000216}/properties/P000162.md (97%) diff --git a/spaces/S000214/README.md b/spaces/S000216/README.md similarity index 98% rename from spaces/S000214/README.md rename to spaces/S000216/README.md index 4c29852258..a7ce6fa464 100644 --- a/spaces/S000214/README.md +++ b/spaces/S000216/README.md @@ -1,5 +1,5 @@ --- -uid: S000214 +uid: S000216 name: Katětov's non-normal subspace of $\beta\mathbb{N}$ refs: - doi: 10.1007/978-1-4615-7819-2 @@ -11,5 +11,3 @@ Constructed in exercise 6Q of {{doi:10.1007/978-1-4615-7819-2}}. Fix a bijection Let $E'$ be the set of limit points of $E$ in {S108}. Then $E'\neq \emptyset$ and $E_1'\cap E_2' = \emptyset$ for $E, E_1, E_2\in\mathcal{E}$. For each $E\in\mathcal{E}$ take $p_E\in E'$ and let $\Pi = \mathbb{N}\cup D$ where $D = \{p_E : E\in\mathcal{E}\}$. Katětov's non-normal subspace of $\beta\mathbb{N}$ is the space $\Pi$. - - diff --git a/spaces/S000214/properties/P000006.md b/spaces/S000216/properties/P000006.md similarity index 83% rename from spaces/S000214/properties/P000006.md rename to spaces/S000216/properties/P000006.md index c241b2e366..d1f7a75b52 100644 --- a/spaces/S000214/properties/P000006.md +++ b/spaces/S000216/properties/P000006.md @@ -1,5 +1,5 @@ --- -space: S000214 +space: S000216 property: P000006 value: true --- diff --git a/spaces/S000214/properties/P000007.md b/spaces/S000216/properties/P000007.md similarity index 95% rename from spaces/S000214/properties/P000007.md rename to spaces/S000216/properties/P000007.md index 9ccc85834a..fc15abac93 100644 --- a/spaces/S000214/properties/P000007.md +++ b/spaces/S000216/properties/P000007.md @@ -1,5 +1,5 @@ --- -space: S000214 +space: S000216 property: P000007 value: false --- diff --git a/spaces/S000214/properties/P000049.md b/spaces/S000216/properties/P000049.md similarity index 94% rename from spaces/S000214/properties/P000049.md rename to spaces/S000216/properties/P000049.md index 2bad15a8b0..e839a0048b 100644 --- a/spaces/S000214/properties/P000049.md +++ b/spaces/S000216/properties/P000049.md @@ -1,5 +1,5 @@ --- -space: S000214 +space: S000216 property: P000049 value: true refs: diff --git a/spaces/S000214/properties/P000162.md b/spaces/S000216/properties/P000162.md similarity index 97% rename from spaces/S000214/properties/P000162.md rename to spaces/S000216/properties/P000162.md index 766ce02181..a8e27711b3 100644 --- a/spaces/S000214/properties/P000162.md +++ b/spaces/S000216/properties/P000162.md @@ -1,5 +1,5 @@ --- -space: S000214 +space: S000216 property: P000162 value: true refs: From b4181caaefa02eeec87b7ff1a471595a360ea718 Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Thu, 25 Dec 2025 21:28:34 +0100 Subject: [PATCH 04/14] update P49 --- spaces/S000216/properties/P000049.md | 5 +---- 1 file changed, 1 insertion(+), 4 deletions(-) diff --git a/spaces/S000216/properties/P000049.md b/spaces/S000216/properties/P000049.md index e839a0048b..6621f07901 100644 --- a/spaces/S000216/properties/P000049.md +++ b/spaces/S000216/properties/P000049.md @@ -2,9 +2,6 @@ space: S000216 property: P000049 value: true -refs: -- zb: "0684.54001" - name: General Topology (Engelking, 1989) --- -It's a dense subspace of {S108} and {S108|P49}. A dense subspace of extremally disconnected space is extremally disconnected (see {{zb:"0684.54001"}} exercise 6.2.G.c). \ No newline at end of file +$X$ is a dense subspace of {S108} and {S108|P49}. \ No newline at end of file From 28b9d751197c7fc3506648480e4b1a3cfa0f4eb4 Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Thu, 25 Dec 2025 21:31:31 +0100 Subject: [PATCH 05/14] deleted P6 --- spaces/S000216/properties/P000006.md | 7 ------- 1 file changed, 7 deletions(-) delete mode 100644 spaces/S000216/properties/P000006.md diff --git a/spaces/S000216/properties/P000006.md b/spaces/S000216/properties/P000006.md deleted file mode 100644 index d1f7a75b52..0000000000 --- a/spaces/S000216/properties/P000006.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000216 -property: P000006 -value: true ---- - -Is a subspace of {S108} and {S108|P6}. From 4995a3b8ed56f7b3ea02d7f05b29d058327d46f2 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Fri, 2 Jan 2026 06:04:38 +0100 Subject: [PATCH 06/14] Update spaces/S000216/README.md Co-authored-by: Felix Pernegger --- spaces/S000216/README.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/spaces/S000216/README.md b/spaces/S000216/README.md index a7ce6fa464..a59368a28a 100644 --- a/spaces/S000216/README.md +++ b/spaces/S000216/README.md @@ -6,8 +6,8 @@ refs: name: Rings of Continuous Functions (Gillman & Jerison) --- -Constructed in exercise 6Q of {{doi:10.1007/978-1-4615-7819-2}}. Fix a bijection $\varphi:\mathbb{N}\to\mathbb{Q}$. For each irrational $r$ fix a sequence of rational numbers $s_n\to r$, and let $E_r = \{\varphi^{-1}(s_n) : n\in\mathbb{N}\}$. Let $\mathcal{E} = \{E_r : r\in\mathbb{R}\setminus\mathbb{Q}\}$. Then $\mathcal{E}$ is an almost disjoint family on $\mathbb{N}$ of size $\mathfrak{c}$. +Fix a bijection $\varphi:\mathbb{N}\to\mathbb{Q}$. For each irrational $r$ fix a sequence of rational numbers $s_n\to r$, and let $E_r = \{\varphi^{-1}(s_n) : n\in\mathbb{N}\}$. Let $\mathcal{E} = \{E_r : r\in\mathbb{R}\setminus\mathbb{Q}\}$. Let $E'$ be the set of limit points of $E$ in {S108}. Then $E'\neq \emptyset$ for $E \in\mathcal{E}$. For each $E\in\mathcal{E}$ pick some $p_E\in E'$. -Let $E'$ be the set of limit points of $E$ in {S108}. Then $E'\neq \emptyset$ and $E_1'\cap E_2' = \emptyset$ for $E, E_1, E_2\in\mathcal{E}$. For each $E\in\mathcal{E}$ take $p_E\in E'$ and let $\Pi = \mathbb{N}\cup D$ where $D = \{p_E : E\in\mathcal{E}\}$. +Katětov's non-normal subspace of $\beta\mathbb{N}$ is the space $X=\mathbb{N}\cup D$ where $D = \{p_E : E\in\mathcal{E}\}$. -Katětov's non-normal subspace of $\beta\mathbb{N}$ is the space $\Pi$. +Constructed in exercise 6Q of {{doi:10.1007/978-1-4615-7819-2}}. From 6ba75ed2b6794e9d01ad6a4c685ba6bb98b0c794 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Fri, 2 Jan 2026 06:06:06 +0100 Subject: [PATCH 07/14] Update spaces/S000216/properties/P000049.md Co-authored-by: Felix Pernegger --- spaces/S000216/properties/P000049.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000216/properties/P000049.md b/spaces/S000216/properties/P000049.md index 6621f07901..24cca63487 100644 --- a/spaces/S000216/properties/P000049.md +++ b/spaces/S000216/properties/P000049.md @@ -4,4 +4,4 @@ property: P000049 value: true --- -$X$ is a dense subspace of {S108} and {S108|P49}. \ No newline at end of file +$X$ contains {S2} and is therefore a dense subspace of {S108} and {S108|P49}. \ No newline at end of file From 62e5d43cc6cdb6120cc9b6ae90a5471d39a295ad Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Fri, 2 Jan 2026 08:46:17 +0100 Subject: [PATCH 08/14] updated this space --- spaces/S000216/properties/P000006.md | 7 +++++++ spaces/S000216/properties/P000007.md | 4 +++- spaces/S000216/properties/P000026.md | 7 +++++++ spaces/S000216/properties/P000065.md | 7 +++++++ spaces/S000216/properties/P000112.md | 7 +++++++ spaces/S000216/properties/P000162.md | 10 ---------- 6 files changed, 31 insertions(+), 11 deletions(-) create mode 100644 spaces/S000216/properties/P000006.md create mode 100644 spaces/S000216/properties/P000026.md create mode 100644 spaces/S000216/properties/P000065.md create mode 100644 spaces/S000216/properties/P000112.md delete mode 100644 spaces/S000216/properties/P000162.md diff --git a/spaces/S000216/properties/P000006.md b/spaces/S000216/properties/P000006.md new file mode 100644 index 0000000000..0dff4f1236 --- /dev/null +++ b/spaces/S000216/properties/P000006.md @@ -0,0 +1,7 @@ +--- +space: S000216 +property: P000006 +value: true +--- + +Since {S108|P6}. diff --git a/spaces/S000216/properties/P000007.md b/spaces/S000216/properties/P000007.md index fc15abac93..3f5a233709 100644 --- a/spaces/S000216/properties/P000007.md +++ b/spaces/S000216/properties/P000007.md @@ -4,4 +4,6 @@ property: P000007 value: false --- -$D$ is a closed discrete subspace of $\Pi$ of size $\mathfrak{c}$. Since there is $2^\mathfrak{c}$ continuous real-valued functions on $D$ and at most $\mathfrak{c}$ continuous real-valued functions on $\Pi$, from Tietze extension theorem $\Pi$ cannot be $T_4$. \ No newline at end of file +$D$ is a closed discrete subspace of $X$ of size $\mathfrak{c}$. There is $2^\mathfrak{c}$ continuous real-valued functions on $D$ and at most $\mathfrak{c}$ continuous real-valued functions on $X$ since {S216|P26}. + +If $X$ were $T_4$ then from Tietze extension theorem we would obtain $2^\mathfrak{c} \leq \mathfrak{c}$, contradiction. So $X$ is not $T_4$. \ No newline at end of file diff --git a/spaces/S000216/properties/P000026.md b/spaces/S000216/properties/P000026.md new file mode 100644 index 0000000000..93feb8b3b0 --- /dev/null +++ b/spaces/S000216/properties/P000026.md @@ -0,0 +1,7 @@ +--- +space: S000216 +property: P000026 +value: true +--- + +$\mathbb{N}\subseteq X$ is countable and dense. \ No newline at end of file diff --git a/spaces/S000216/properties/P000065.md b/spaces/S000216/properties/P000065.md new file mode 100644 index 0000000000..7ca5f419ff --- /dev/null +++ b/spaces/S000216/properties/P000065.md @@ -0,0 +1,7 @@ +--- +space: S000216 +property: P000065 +value: true +--- + +$X$ is in bijection with $\mathbb{R}$ diff --git a/spaces/S000216/properties/P000112.md b/spaces/S000216/properties/P000112.md new file mode 100644 index 0000000000..54c76c678b --- /dev/null +++ b/spaces/S000216/properties/P000112.md @@ -0,0 +1,7 @@ +--- +space: S000216 +property: P000112 +value: true +--- + +Extend $\varphi$ to $X$ so that if $E = E_r = \{\varphi^{-1}(s_n) : n\in\omega\}$ and $s_n\to r$ then $\varphi(p_E) = r$. If $\varphi(p_E)\in U$ where $U\subseteq \mathbb{R}$ is open, find $N$ such that $s_n\in U$ for $n\geq N$, then $V =\overline{E}\setminus\varphi^{-1}(\{s_1, s_2, ..., s_N\})$ is an open neighbourhood of $p_E$ and $\varphi(V)\subseteq U$. So $\varphi:X\to\mathbb{R}$ is a continuous injection, hence $X$ is submetrizable. diff --git a/spaces/S000216/properties/P000162.md b/spaces/S000216/properties/P000162.md deleted file mode 100644 index a8e27711b3..0000000000 --- a/spaces/S000216/properties/P000162.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000216 -property: P000162 -value: true -refs: - - doi: 10.1007/978-1-4615-7819-2 - name: Rings of Continuous Functions (Gillman & Jerison) ---- - -Extend $\varphi$ to $\Pi$ so that if $E = E_r$ and $s_n\to r$ then $\varphi(p_E) = \lim_{n\to\infty} \varphi(s_n)$. If $\varphi(p_E)\in U$ where $U\subseteq \mathbb{R}$ is open, find $N$ such that $s_n\in U$ for $n\geq N$, then $V =\overline{E}\setminus\varphi^{-1}(\{s_1, s_2, ..., s_N\})$ is an open neighbourhood of $p_E$ and $\varphi(V)\subseteq U$. This shows that $\varphi:\Pi\to\mathbb{R}$ is continuous. By definition it's clearly a bijection. From corollary 8.18 of {{10.1007/978-1-4615-7819-2}}, $\Pi$ is realcompact. \ No newline at end of file From 8f6828684cf5eb7f1ad9d3f9ece058e64cc0454d Mon Sep 17 00:00:00 2001 From: Batixx Date: Fri, 2 Jan 2026 12:30:48 +0100 Subject: [PATCH 09/14] add missing newlines --- spaces/S000216/properties/P000007.md | 2 +- spaces/S000216/properties/P000026.md | 2 +- spaces/S000216/properties/P000049.md | 2 +- 3 files changed, 3 insertions(+), 3 deletions(-) diff --git a/spaces/S000216/properties/P000007.md b/spaces/S000216/properties/P000007.md index 3f5a233709..56369b9d8f 100644 --- a/spaces/S000216/properties/P000007.md +++ b/spaces/S000216/properties/P000007.md @@ -6,4 +6,4 @@ value: false $D$ is a closed discrete subspace of $X$ of size $\mathfrak{c}$. There is $2^\mathfrak{c}$ continuous real-valued functions on $D$ and at most $\mathfrak{c}$ continuous real-valued functions on $X$ since {S216|P26}. -If $X$ were $T_4$ then from Tietze extension theorem we would obtain $2^\mathfrak{c} \leq \mathfrak{c}$, contradiction. So $X$ is not $T_4$. \ No newline at end of file +If $X$ were $T_4$ then from Tietze extension theorem we would obtain $2^\mathfrak{c} \leq \mathfrak{c}$, contradiction. So $X$ is not $T_4$. diff --git a/spaces/S000216/properties/P000026.md b/spaces/S000216/properties/P000026.md index 93feb8b3b0..89e74fe744 100644 --- a/spaces/S000216/properties/P000026.md +++ b/spaces/S000216/properties/P000026.md @@ -4,4 +4,4 @@ property: P000026 value: true --- -$\mathbb{N}\subseteq X$ is countable and dense. \ No newline at end of file +$\mathbb{N}\subseteq X$ is countable and dense. diff --git a/spaces/S000216/properties/P000049.md b/spaces/S000216/properties/P000049.md index 24cca63487..5356418517 100644 --- a/spaces/S000216/properties/P000049.md +++ b/spaces/S000216/properties/P000049.md @@ -4,4 +4,4 @@ property: P000049 value: true --- -$X$ contains {S2} and is therefore a dense subspace of {S108} and {S108|P49}. \ No newline at end of file +$X$ contains {S2} and is therefore a dense subspace of {S108} and {S108|P49}. From 16bc66307aac580833aab05df94dd702ca7012a5 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Fri, 2 Jan 2026 12:59:47 +0100 Subject: [PATCH 10/14] Update spaces/S000216/properties/P000006.md Co-authored-by: Felix Pernegger --- spaces/S000216/properties/P000006.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000216/properties/P000006.md b/spaces/S000216/properties/P000006.md index 0dff4f1236..911be328b0 100644 --- a/spaces/S000216/properties/P000006.md +++ b/spaces/S000216/properties/P000006.md @@ -4,4 +4,4 @@ property: P000006 value: true --- -Since {S108|P6}. +$X$ is contained in {S108} and {S108|P6}. From 29b016b33997847e049948d92e895d1fb2282763 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Fri, 2 Jan 2026 13:00:57 +0100 Subject: [PATCH 11/14] Update spaces/S000216/properties/P000065.md Co-authored-by: Felix Pernegger --- spaces/S000216/properties/P000065.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000216/properties/P000065.md b/spaces/S000216/properties/P000065.md index 7ca5f419ff..7276c66117 100644 --- a/spaces/S000216/properties/P000065.md +++ b/spaces/S000216/properties/P000065.md @@ -4,4 +4,4 @@ property: P000065 value: true --- -$X$ is in bijection with $\mathbb{R}$ +$X$ is in bijection with $\mathbb{R}$. From ab3f5c7656912cef74447cae546a6d12fe5ac69f Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Fri, 2 Jan 2026 13:01:30 +0100 Subject: [PATCH 12/14] Update spaces/S000216/README.md Co-authored-by: Felix Pernegger --- spaces/S000216/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000216/README.md b/spaces/S000216/README.md index a59368a28a..87ada22090 100644 --- a/spaces/S000216/README.md +++ b/spaces/S000216/README.md @@ -6,7 +6,7 @@ refs: name: Rings of Continuous Functions (Gillman & Jerison) --- -Fix a bijection $\varphi:\mathbb{N}\to\mathbb{Q}$. For each irrational $r$ fix a sequence of rational numbers $s_n\to r$, and let $E_r = \{\varphi^{-1}(s_n) : n\in\mathbb{N}\}$. Let $\mathcal{E} = \{E_r : r\in\mathbb{R}\setminus\mathbb{Q}\}$. Let $E'$ be the set of limit points of $E$ in {S108}. Then $E'\neq \emptyset$ for $E \in\mathcal{E}$. For each $E\in\mathcal{E}$ pick some $p_E\in E'$. +Fix a bijection $\varphi:\mathbb{N}\to\mathbb{Q}$. For each irrational $r$ fix a sequence of rational numbers $s_n\to r$, and let $E_r = \{\varphi^{-1}(s_n) : n\in\mathbb{N}\}$. Let $\mathcal{E} = \{E_r : r\in\mathbb{R}\setminus\mathbb{Q}\}$. Let $E'$ be the set of limit points for a subset $E$ of {S108}. Then $E'\neq \emptyset$ for $E \in\mathcal{E}$. For each $E\in\mathcal{E}$ pick some $p_E\in E'$. Katětov's non-normal subspace of $\beta\mathbb{N}$ is the space $X=\mathbb{N}\cup D$ where $D = \{p_E : E\in\mathcal{E}\}$. From e7650bcdca974c363402a35fe7b7b84b6eebd720 Mon Sep 17 00:00:00 2001 From: Moniker1998 <88507423+Moniker1998@users.noreply.github.com> Date: Fri, 2 Jan 2026 13:02:40 +0100 Subject: [PATCH 13/14] Update spaces/S000216/properties/P000112.md Co-authored-by: Felix Pernegger --- spaces/S000216/properties/P000112.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000216/properties/P000112.md b/spaces/S000216/properties/P000112.md index 54c76c678b..2166e2d2f0 100644 --- a/spaces/S000216/properties/P000112.md +++ b/spaces/S000216/properties/P000112.md @@ -4,4 +4,4 @@ property: P000112 value: true --- -Extend $\varphi$ to $X$ so that if $E = E_r = \{\varphi^{-1}(s_n) : n\in\omega\}$ and $s_n\to r$ then $\varphi(p_E) = r$. If $\varphi(p_E)\in U$ where $U\subseteq \mathbb{R}$ is open, find $N$ such that $s_n\in U$ for $n\geq N$, then $V =\overline{E}\setminus\varphi^{-1}(\{s_1, s_2, ..., s_N\})$ is an open neighbourhood of $p_E$ and $\varphi(V)\subseteq U$. So $\varphi:X\to\mathbb{R}$ is a continuous injection, hence $X$ is submetrizable. +Extend $\varphi$ to $X$ so that if $E = E_r = \{\varphi^{-1}(s_n) : n\in\omega\}$ and $s_n\to r$ then $\varphi(p_E) = r$. If $\varphi(p_E)\in U$ where $U\subseteq \mathbb{R}$ is open, find $N$ such that $s_n\in U$ for $n\geq N$. Since {S216|P49}, $V =\overline{E}\setminus\varphi^{-1}(\{s_1, s_2, ..., s_N\})$ is an open neighbourhood of $p_E$ and $\varphi(V)\subseteq U$. So $\varphi:X\to\mathbb{R}$ is a continuous injection, hence $X$ is submetrizable. From b1b038519a1b27d46313a171a2c3cb0f4dfbb9c9 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Fri, 2 Jan 2026 13:18:54 +0100 Subject: [PATCH 14/14] Apply suggestion from @felixpernegger --- spaces/S000216/properties/P000007.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000216/properties/P000007.md b/spaces/S000216/properties/P000007.md index 56369b9d8f..d5683934f0 100644 --- a/spaces/S000216/properties/P000007.md +++ b/spaces/S000216/properties/P000007.md @@ -4,6 +4,6 @@ property: P000007 value: false --- -$D$ is a closed discrete subspace of $X$ of size $\mathfrak{c}$. There is $2^\mathfrak{c}$ continuous real-valued functions on $D$ and at most $\mathfrak{c}$ continuous real-valued functions on $X$ since {S216|P26}. +$D$ is a closed discrete subspace of $X$ of size $\mathfrak{c}$, so there are $2^\mathfrak{c}$ continuous real-valued functions on $D$ and at most $\mathfrak{c}$ continuous real-valued functions on $X$ since {S216|P26}. If $X$ were $T_4$ then from Tietze extension theorem we would obtain $2^\mathfrak{c} \leq \mathfrak{c}$, contradiction. So $X$ is not $T_4$.