From 667ce86d4a683796fc8ddc9b1503b63cffea274a Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Sun, 11 Jan 2026 08:23:09 +0800 Subject: [PATCH 01/10] =?UTF-8?q?Has=20closed=20discrete=20subset=20of=20s?= =?UTF-8?q?ize=20=F0=9D=94=A0=20(part=202)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- properties/P000227.md | 5 +++++ spaces/S000009/properties/P000198.md | 7 ------- spaces/S000009/properties/P000227.md | 7 +++++++ spaces/S000057/properties/P000013.md | 12 ------------ spaces/S000057/properties/P000227.md | 7 +++++++ spaces/S000060/properties/P000227.md | 7 +++++++ spaces/S000063/properties/{P000198.md => P000227.md} | 6 ++++-- spaces/S000064/properties/P000227.md | 7 +++++++ spaces/S000068/properties/P000198.md | 7 ------- spaces/S000068/properties/P000227.md | 7 +++++++ spaces/S000070/properties/P000227.md | 7 +++++++ spaces/S000074/properties/P000227.md | 7 +++++++ spaces/S000076/properties/P000227.md | 7 +++++++ spaces/S000085/properties/{P000198.md => P000227.md} | 5 +++-- spaces/S000086/properties/P000198.md | 7 ------- spaces/S000086/properties/P000227.md | 7 +++++++ spaces/S000102/properties/P000227.md | 7 +++++++ spaces/S000110/properties/{P000198.md => P000227.md} | 4 ++-- spaces/S000129/properties/P000198.md | 7 ------- spaces/S000129/properties/P000227.md | 7 +++++++ spaces/S000133/properties/P000227.md | 7 +++++++ spaces/S000134/properties/{P000198.md => P000227.md} | 4 ++-- spaces/S000135/properties/{P000198.md => P000227.md} | 6 +++--- spaces/S000175/properties/P000198.md | 10 ---------- spaces/S000175/properties/P000227.md | 10 ++++++++++ spaces/S000182/properties/P000227.md | 7 +++++++ 26 files changed, 120 insertions(+), 61 deletions(-) delete mode 100644 spaces/S000009/properties/P000198.md create mode 100644 spaces/S000009/properties/P000227.md delete mode 100644 spaces/S000057/properties/P000013.md create mode 100644 spaces/S000057/properties/P000227.md create mode 100644 spaces/S000060/properties/P000227.md rename spaces/S000063/properties/{P000198.md => P000227.md} (63%) create mode 100644 spaces/S000064/properties/P000227.md delete mode 100644 spaces/S000068/properties/P000198.md create mode 100644 spaces/S000068/properties/P000227.md create mode 100644 spaces/S000070/properties/P000227.md create mode 100644 spaces/S000074/properties/P000227.md create mode 100644 spaces/S000076/properties/P000227.md rename spaces/S000085/properties/{P000198.md => P000227.md} (77%) delete mode 100644 spaces/S000086/properties/P000198.md create mode 100644 spaces/S000086/properties/P000227.md create mode 100644 spaces/S000102/properties/P000227.md rename spaces/S000110/properties/{P000198.md => P000227.md} (84%) delete mode 100644 spaces/S000129/properties/P000198.md create mode 100644 spaces/S000129/properties/P000227.md create mode 100644 spaces/S000133/properties/P000227.md rename spaces/S000134/properties/{P000198.md => P000227.md} (87%) rename spaces/S000135/properties/{P000198.md => P000227.md} (78%) delete mode 100644 spaces/S000175/properties/P000198.md create mode 100644 spaces/S000175/properties/P000227.md create mode 100644 spaces/S000182/properties/P000227.md diff --git a/properties/P000227.md b/properties/P000227.md index 78b5375c55..75fe9c574e 100644 --- a/properties/P000227.md +++ b/properties/P000227.md @@ -16,3 +16,8 @@ Compare with these properties, where $D$ denotes a discrete closed set in $X$: - {P107} $(\exists D: |D|=1)$ - {P21} $(\forall D: |D|<\aleph_0)$ - {P198} $(\forall D: |D|\le\aleph_0)$ + +---- +#### Meta-properties + +- This property is preserved in any finer topology. diff --git a/spaces/S000009/properties/P000198.md b/spaces/S000009/properties/P000198.md deleted file mode 100644 index 5e0a4416d2..0000000000 --- a/spaces/S000009/properties/P000198.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000009 -property: P000198 -value: false ---- - -Let $p$ be the particular point. $X\setminus\{p\}$ is an uncountable, closed and discrete subspace. diff --git a/spaces/S000009/properties/P000227.md b/spaces/S000009/properties/P000227.md new file mode 100644 index 0000000000..119fa4e2dc --- /dev/null +++ b/spaces/S000009/properties/P000227.md @@ -0,0 +1,7 @@ +--- +space: S000009 +property: P000227 +value: true +--- + +Let $p$ be the particular point. $X\setminus\{p\}$ is an closed and discrete subspace of cardinality $\mathfrak c$. diff --git a/spaces/S000057/properties/P000013.md b/spaces/S000057/properties/P000013.md deleted file mode 100644 index 573a2953d5..0000000000 --- a/spaces/S000057/properties/P000013.md +++ /dev/null @@ -1,12 +0,0 @@ ---- -space: S000057 -property: P000013 -value: false -refs: -- zb: "0205.26601" - name: General topology (S. Willard) ---- - -{S57|P26} and -$X\setminus\mathbb Q$ is closed and discrete subset of cardinality continuum. -Therefore the space is not {P13}, as shown by *Jones' lemma* (Lemma 15.2 in {{zb:0205.26601}}). diff --git a/spaces/S000057/properties/P000227.md b/spaces/S000057/properties/P000227.md new file mode 100644 index 0000000000..8945fd5247 --- /dev/null +++ b/spaces/S000057/properties/P000227.md @@ -0,0 +1,7 @@ +--- +space: S000057 +property: P000227 +value: true +--- + +$X \setminus \mathbb Q$ is a closed and discrete subset of cardinality continuum. diff --git a/spaces/S000060/properties/P000227.md b/spaces/S000060/properties/P000227.md new file mode 100644 index 0000000000..647e3686ea --- /dev/null +++ b/spaces/S000060/properties/P000227.md @@ -0,0 +1,7 @@ +--- +space: S000060 +property: P000227 +value: true +--- + +The subspace $\mathbb R\setminus\mathbb Q$ is closed discrete with cardinality $\mathfrak c$. diff --git a/spaces/S000063/properties/P000198.md b/spaces/S000063/properties/P000227.md similarity index 63% rename from spaces/S000063/properties/P000198.md rename to spaces/S000063/properties/P000227.md index 54a0a6758b..56fc7370dd 100644 --- a/spaces/S000063/properties/P000198.md +++ b/spaces/S000063/properties/P000227.md @@ -1,7 +1,7 @@ --- space: S000063 -property: P000198 -value: false +property: P000227 +value: true --- Let $\{q_n: n\in\mathbb N\}=\mathbb Q$. @@ -9,3 +9,5 @@ Consider the closed set $L = \mathbb R\setminus \bigcup_{n\in \mathbb N} (q_n-2^{-n},q_n+2^{-n})$. It is discrete, as a subset of the irrational numbers. Since the Lebesque measure of $L$ is infinite (its complement is of finite measure), $L$ is uncountable. + +As $L$ is also closed in Euclidean topology, any uncountable closed set in Euclidean topology is of cardinality $\mathfrak c$, by Cantor–Bendixson theorem. diff --git a/spaces/S000064/properties/P000227.md b/spaces/S000064/properties/P000227.md new file mode 100644 index 0000000000..a62e498a65 --- /dev/null +++ b/spaces/S000064/properties/P000227.md @@ -0,0 +1,7 @@ +--- +space: S000064 +property: P000227 +value: true +--- + +The subspace $X \setminus D$ is closed discrete with cardinality $\mathfrak c$. diff --git a/spaces/S000068/properties/P000198.md b/spaces/S000068/properties/P000198.md deleted file mode 100644 index 2c463eb4ed..0000000000 --- a/spaces/S000068/properties/P000198.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000068 -property: P000198 -value: false ---- - -The subspace $\mathbb R\times\{a\}$ is closed and discrete for any $a\in\mathbb R$. diff --git a/spaces/S000068/properties/P000227.md b/spaces/S000068/properties/P000227.md new file mode 100644 index 0000000000..1e012e9816 --- /dev/null +++ b/spaces/S000068/properties/P000227.md @@ -0,0 +1,7 @@ +--- +space: S000068 +property: P000227 +value: true +--- + +The subspace $\mathbb R\times\{0\}$ is closed and discrete with cardinality $\mathfrak c$. diff --git a/spaces/S000070/properties/P000227.md b/spaces/S000070/properties/P000227.md new file mode 100644 index 0000000000..e5c4e172cb --- /dev/null +++ b/spaces/S000070/properties/P000227.md @@ -0,0 +1,7 @@ +--- +space: S000070 +property: P000227 +value: true +--- + +The subspace $L$ is closed and discrete with cardinality $\mathfrak c$. diff --git a/spaces/S000074/properties/P000227.md b/spaces/S000074/properties/P000227.md new file mode 100644 index 0000000000..e8ff1c6f61 --- /dev/null +++ b/spaces/S000074/properties/P000227.md @@ -0,0 +1,7 @@ +--- +space: S000074 +property: P000227 +value: true +--- + +The subspace $\mathbb R\times\{0\}$ is closed and discrete with cardinality $\mathfrak c$. diff --git a/spaces/S000076/properties/P000227.md b/spaces/S000076/properties/P000227.md new file mode 100644 index 0000000000..68c3150abd --- /dev/null +++ b/spaces/S000076/properties/P000227.md @@ -0,0 +1,7 @@ +--- +space: S000076 +property: P000227 +value: true +--- + +The closed set $\{(x,-x):x\in\mathbb R\}$ is a discrete subspace with cardinality $\mathfrak c$. diff --git a/spaces/S000085/properties/P000198.md b/spaces/S000085/properties/P000227.md similarity index 77% rename from spaces/S000085/properties/P000198.md rename to spaces/S000085/properties/P000227.md index 8e47331808..41c0277452 100644 --- a/spaces/S000085/properties/P000198.md +++ b/spaces/S000085/properties/P000227.md @@ -1,6 +1,7 @@ --- space: S000085 -property: P000198 -value: false +property: P000227 +value: true --- + The set of origins is a closed and discrete subspace of cardinality $2^\mathfrak c$. diff --git a/spaces/S000086/properties/P000198.md b/spaces/S000086/properties/P000198.md deleted file mode 100644 index 05dc4448f8..0000000000 --- a/spaces/S000086/properties/P000198.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000086 -property: P000198 -value: false ---- - -The subspace $\mathbb{R}{\times}\{1\}$ is closed uncountable and discrete. diff --git a/spaces/S000086/properties/P000227.md b/spaces/S000086/properties/P000227.md new file mode 100644 index 0000000000..f9b1448882 --- /dev/null +++ b/spaces/S000086/properties/P000227.md @@ -0,0 +1,7 @@ +--- +space: S000086 +property: P000227 +value: true +--- + +The subspace $\mathbb{R}{\times}\{1\}$ is closed and discrete with cardinality $\mathfrak c$. diff --git a/spaces/S000102/properties/P000227.md b/spaces/S000102/properties/P000227.md new file mode 100644 index 0000000000..29def7d87a --- /dev/null +++ b/spaces/S000102/properties/P000227.md @@ -0,0 +1,7 @@ +--- +space: S000102 +property: P000227 +value: true +--- + +The subspace $\mathbb R \times \{0\}^\omega$ is closed discrete with cardinality $\mathfrak c$. diff --git a/spaces/S000110/properties/P000198.md b/spaces/S000110/properties/P000227.md similarity index 84% rename from spaces/S000110/properties/P000198.md rename to spaces/S000110/properties/P000227.md index 88b15ad805..12e1de92ff 100644 --- a/spaces/S000110/properties/P000198.md +++ b/spaces/S000110/properties/P000227.md @@ -1,7 +1,7 @@ --- space: S000110 -property: P000198 -value: false +property: P000227 +value: true --- By construction, $M$ is a closed discrete subspace of $X$. There are $2^\mathfrak{c}$ free ultrafilters on $\omega$, i.e., $|M| = 2^\mathfrak{c}$. diff --git a/spaces/S000129/properties/P000198.md b/spaces/S000129/properties/P000198.md deleted file mode 100644 index 0cd05ae324..0000000000 --- a/spaces/S000129/properties/P000198.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000129 -property: P000198 -value: false ---- - -$A=\{x\in X: \|x\|=1/2\}$ is a closed discrete and uncountable subset of $X$. diff --git a/spaces/S000129/properties/P000227.md b/spaces/S000129/properties/P000227.md new file mode 100644 index 0000000000..b698f5f7c8 --- /dev/null +++ b/spaces/S000129/properties/P000227.md @@ -0,0 +1,7 @@ +--- +space: S000129 +property: P000227 +value: true +--- + +$A=\{x\in X: \|x\|=1/2\}$ is a closed discrete subset of $X$ of cardinality $\mathfrak c$. diff --git a/spaces/S000133/properties/P000227.md b/spaces/S000133/properties/P000227.md new file mode 100644 index 0000000000..41ea57e57f --- /dev/null +++ b/spaces/S000133/properties/P000227.md @@ -0,0 +1,7 @@ +--- +space: S000133 +property: P000227 +value: true +--- + +The subspace $V_0 \setminus V_1$ is closed discrete with cardinality $\mathfrak c$. diff --git a/spaces/S000134/properties/P000198.md b/spaces/S000134/properties/P000227.md similarity index 87% rename from spaces/S000134/properties/P000198.md rename to spaces/S000134/properties/P000227.md index c83b9f30bb..b8e6f69da9 100644 --- a/spaces/S000134/properties/P000198.md +++ b/spaces/S000134/properties/P000227.md @@ -1,7 +1,7 @@ --- space: S000134 -property: P000198 -value: false +property: P000227 +value: true --- Every circle $C_a=\{(x,y)\mid x^2+y^2=a^2\}$ is closed, as the space is finer than the {S176}, and discrete, as balls $B_\epsilon(x)$ contain at most one point of $C_a$ whenever $\epsilon\le a$. diff --git a/spaces/S000135/properties/P000198.md b/spaces/S000135/properties/P000227.md similarity index 78% rename from spaces/S000135/properties/P000198.md rename to spaces/S000135/properties/P000227.md index d99ac4252f..133373bb0d 100644 --- a/spaces/S000135/properties/P000198.md +++ b/spaces/S000135/properties/P000227.md @@ -1,7 +1,7 @@ --- space: S000135 -property: P000198 -value: false +property: P000227 +value: true refs: - zb: "0386.54001" name: Counterexamples in Topology @@ -9,4 +9,4 @@ refs: This topology is finer than {S134}, (cf. item #1 for space #141 in {{zb:0386.54001}}) -and {S134|P198}. +and {S134|P227}. diff --git a/spaces/S000175/properties/P000198.md b/spaces/S000175/properties/P000198.md deleted file mode 100644 index 97d9d62f1a..0000000000 --- a/spaces/S000175/properties/P000198.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000175 -property: P000198 -value: false -refs: - - doi: 10.2307/2315929 - name: Problem 5468 of The American Mathematical Monthly ---- - -Contains a closed discrete uncountable subspace $\{(x,y):x^2+y^2=1\}$. diff --git a/spaces/S000175/properties/P000227.md b/spaces/S000175/properties/P000227.md new file mode 100644 index 0000000000..2c3487e2f0 --- /dev/null +++ b/spaces/S000175/properties/P000227.md @@ -0,0 +1,10 @@ +--- +space: S000175 +property: P000227 +value: true +refs: + - doi: 10.2307/2315929 + name: Problem 5468 of The American Mathematical Monthly +--- + +Contains a closed discrete subspace $\{(x,y):x^2+y^2=1\}$ of cardinality $\mathfrak c$. diff --git a/spaces/S000182/properties/P000227.md b/spaces/S000182/properties/P000227.md new file mode 100644 index 0000000000..5498d9d0eb --- /dev/null +++ b/spaces/S000182/properties/P000227.md @@ -0,0 +1,7 @@ +--- +space: S000182 +property: P000227 +value: true +--- + +The subspace $D \times \{0\}$ is closed discrete with cardinality $\mathfrak c$. From 8163840b664c5375c8a0c62bc0cc7a31ea9c93bc Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Sun, 11 Jan 2026 08:36:37 +0800 Subject: [PATCH 02/10] More spaces --- spaces/S000070/properties/P000198.md | 11 ----------- spaces/S000074/properties/P000198.md | 11 ----------- spaces/S000076/properties/P000198.md | 12 ------------ spaces/S000091/properties/P000198.md | 7 ------- spaces/S000091/properties/P000227.md | 7 +++++++ spaces/S000214/properties/{P000198.md => P000227.md} | 4 ++-- 6 files changed, 9 insertions(+), 43 deletions(-) delete mode 100644 spaces/S000070/properties/P000198.md delete mode 100644 spaces/S000074/properties/P000198.md delete mode 100644 spaces/S000076/properties/P000198.md delete mode 100644 spaces/S000091/properties/P000198.md create mode 100644 spaces/S000091/properties/P000227.md rename spaces/S000214/properties/{P000198.md => P000227.md} (82%) diff --git a/spaces/S000070/properties/P000198.md b/spaces/S000070/properties/P000198.md deleted file mode 100644 index e1df4df48d..0000000000 --- a/spaces/S000070/properties/P000198.md +++ /dev/null @@ -1,11 +0,0 @@ ---- -space: S000070 -property: P000198 -value: false -refs: -- zb: "0386.54001" - name: Counterexamples in Topology ---- -The subspace $L$ is closed and discrete. - -See item #5 for space #78 in {{zb:0386.54001}}. diff --git a/spaces/S000074/properties/P000198.md b/spaces/S000074/properties/P000198.md deleted file mode 100644 index 4473f9d0af..0000000000 --- a/spaces/S000074/properties/P000198.md +++ /dev/null @@ -1,11 +0,0 @@ ---- -space: S000074 -property: P000198 -value: false -refs: -- zb: "0386.54001" - name: Counterexamples in Topology ---- -The subspace $\mathbb R\times \{0\}$ is closed and discrete. - -See item #4 for space #82 in {{zb:0386.54001}}. diff --git a/spaces/S000076/properties/P000198.md b/spaces/S000076/properties/P000198.md deleted file mode 100644 index ce7af995e8..0000000000 --- a/spaces/S000076/properties/P000198.md +++ /dev/null @@ -1,12 +0,0 @@ ---- -space: S000076 -property: P000198 -value: false -refs: -- zb: "0386.54001" - name: Counterexamples in Topology ---- - -The uncountable and closed set $\{(x,-x):x\in\mathbb R\}$ is a discrete subspace. - -See item #2 for space #84 in {{zb:0386.54001}}. \ No newline at end of file diff --git a/spaces/S000091/properties/P000198.md b/spaces/S000091/properties/P000198.md deleted file mode 100644 index 7c1ae658e9..0000000000 --- a/spaces/S000091/properties/P000198.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000091 -property: P000198 -value: false ---- - -The subset $L_0$ is uncountable closed and discrete. diff --git a/spaces/S000091/properties/P000227.md b/spaces/S000091/properties/P000227.md new file mode 100644 index 0000000000..a977d520c5 --- /dev/null +++ b/spaces/S000091/properties/P000227.md @@ -0,0 +1,7 @@ +--- +space: S000091 +property: P000227 +value: true +--- + +The subspace $L_0$ is closed and discrete with cardinality $\mathfrak c$. diff --git a/spaces/S000214/properties/P000198.md b/spaces/S000214/properties/P000227.md similarity index 82% rename from spaces/S000214/properties/P000198.md rename to spaces/S000214/properties/P000227.md index 94c5be0053..cd843d7ff6 100644 --- a/spaces/S000214/properties/P000198.md +++ b/spaces/S000214/properties/P000227.md @@ -1,7 +1,7 @@ --- space: S000214 -property: P000198 -value: false +property: P000227 +value: true --- The subset $\{f\in X: f(\lambda)=0\text{ for }\lambda\in[\omega,\omega_1)\}$ From 391a0c2656f9ec50121d43c58540979250126b2f Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Sun, 11 Jan 2026 08:41:06 +0800 Subject: [PATCH 03/10] and S216. --- spaces/S000216/properties/P000007.md | 9 --------- spaces/S000216/properties/P000227.md | 7 +++++++ 2 files changed, 7 insertions(+), 9 deletions(-) delete mode 100644 spaces/S000216/properties/P000007.md create mode 100644 spaces/S000216/properties/P000227.md diff --git a/spaces/S000216/properties/P000007.md b/spaces/S000216/properties/P000007.md deleted file mode 100644 index d5683934f0..0000000000 --- a/spaces/S000216/properties/P000007.md +++ /dev/null @@ -1,9 +0,0 @@ ---- -space: S000216 -property: P000007 -value: false ---- - -$D$ 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 $T_4$. diff --git a/spaces/S000216/properties/P000227.md b/spaces/S000216/properties/P000227.md new file mode 100644 index 0000000000..36f1087be9 --- /dev/null +++ b/spaces/S000216/properties/P000227.md @@ -0,0 +1,7 @@ +--- +space: S000216 +property: P000227 +value: true +--- + +$D$ is a closed discrete subspace of $X$ of size $\mathfrak{c}$. From 5df92bfc824b89dc6b206c57a11cf57590b87f3d Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Sun, 11 Jan 2026 08:46:28 +0800 Subject: [PATCH 04/10] =?UTF-8?q?Lebesque=20=E2=86=92=20Lebesgue?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- spaces/S000063/properties/P000227.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000063/properties/P000227.md b/spaces/S000063/properties/P000227.md index 56fc7370dd..d2ae34109f 100644 --- a/spaces/S000063/properties/P000227.md +++ b/spaces/S000063/properties/P000227.md @@ -8,6 +8,6 @@ Let $\{q_n: n\in\mathbb N\}=\mathbb Q$. Consider the closed set $L = \mathbb R\setminus \bigcup_{n\in \mathbb N} (q_n-2^{-n},q_n+2^{-n})$. It is discrete, as a subset of the irrational numbers. -Since the Lebesque measure of $L$ is infinite (its complement is of finite measure), $L$ is uncountable. +Since the Lebesgue measure of $L$ is infinite (its complement is of finite measure), $L$ is uncountable. As $L$ is also closed in Euclidean topology, any uncountable closed set in Euclidean topology is of cardinality $\mathfrak c$, by Cantor–Bendixson theorem. From bbde466f135fd921b5f2a872eb8846ac7cb39969 Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Sun, 11 Jan 2026 23:53:57 +0800 Subject: [PATCH 05/10] update description --- properties/P000227.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/properties/P000227.md b/properties/P000227.md index 75fe9c574e..2cf1a11c2b 100644 --- a/properties/P000227.md +++ b/properties/P000227.md @@ -1,18 +1,18 @@ --- uid: P000227 -name: Has a discrete closed set of size $\mathfrak c$ +name: Has a discrete closed subset of size $\mathfrak c$ --- -$X$ has a discrete closed set of cardinality $\mathfrak c=2^{\aleph_0}$. +$X$ has a discrete closed subset of cardinality $\mathfrak c=2^{\aleph_0}$. -*Note*: Discrete closed sets are exactly the sets with no limit point in $X$. +*Note*: Discrete closed subsets are exactly the subsets with no limit point in $X$. *Note*: This property implies $e(X)\ge\mathfrak c$, -where the *extent* $e(X)$ is the supremum of the cardinality of discrete closed sets in $X$. -But there are spaces with $e(X)=\mathfrak c$ and without discrete closed set of cardinality $\mathfrak c$, -i.e., where the supremum is not attained. +where the *extent* $e(X)$ is the supremum of the cardinality of discrete closed subsets in $X$. +But there are spaces with $e(X)=\mathfrak c$ and without discrete closed subset of cardinality $\mathfrak c$, +i.e., where the supremum is not attained, under certain set-theoretical assumptions. -Compare with these properties, where $D$ denotes a discrete closed set in $X$: +Compare with these properties, where $D$ denotes a discrete closed subset in $X$: - {P107} $(\exists D: |D|=1)$ - {P21} $(\forall D: |D|<\aleph_0)$ - {P198} $(\forall D: |D|\le\aleph_0)$ From ee8fdc12c710be21039156825dae66e204af4ab6 Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Mon, 12 Jan 2026 16:17:00 +0800 Subject: [PATCH 06/10] Apply suggestions from code review Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000227.md | 5 +++-- spaces/S000009/properties/P000227.md | 2 +- 2 files changed, 4 insertions(+), 3 deletions(-) diff --git a/properties/P000227.md b/properties/P000227.md index 2cf1a11c2b..509233afe3 100644 --- a/properties/P000227.md +++ b/properties/P000227.md @@ -9,8 +9,9 @@ $X$ has a discrete closed subset of cardinality $\mathfrak c=2^{\aleph_0}$. *Note*: This property implies $e(X)\ge\mathfrak c$, where the *extent* $e(X)$ is the supremum of the cardinality of discrete closed subsets in $X$. -But there are spaces with $e(X)=\mathfrak c$ and without discrete closed subset of cardinality $\mathfrak c$, -i.e., where the supremum is not attained, under certain set-theoretical assumptions. +But, under certain set-theoretic assumptions (for example, if $\mathfrak c=\aleph_{\omega_1}$), +there are spaces with $e(X)=\mathfrak c$ and without discrete closed subset of cardinality $\mathfrak c$, +i.e., where the supremum is not attained. Compare with these properties, where $D$ denotes a discrete closed subset in $X$: - {P107} $(\exists D: |D|=1)$ diff --git a/spaces/S000009/properties/P000227.md b/spaces/S000009/properties/P000227.md index 119fa4e2dc..71d2385a14 100644 --- a/spaces/S000009/properties/P000227.md +++ b/spaces/S000009/properties/P000227.md @@ -4,4 +4,4 @@ property: P000227 value: true --- -Let $p$ be the particular point. $X\setminus\{p\}$ is an closed and discrete subspace of cardinality $\mathfrak c$. +Let $p$ be the particular point. $X\setminus\{p\}$ is a closed and discrete subspace of cardinality $\mathfrak c$. From 56ca10c65af726494e2049b5a33582ee7c08afbd Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Mon, 12 Jan 2026 18:24:01 -0500 Subject: [PATCH 07/10] S63 update --- spaces/S000063/properties/P000227.md | 17 +++++++++++------ 1 file changed, 11 insertions(+), 6 deletions(-) diff --git a/spaces/S000063/properties/P000227.md b/spaces/S000063/properties/P000227.md index d2ae34109f..0c96526e61 100644 --- a/spaces/S000063/properties/P000227.md +++ b/spaces/S000063/properties/P000227.md @@ -2,12 +2,17 @@ space: S000063 property: P000227 value: true +refs: +- mathse: 740095 + name: "Bijection between closed uncountable subset of $\mathbb R$ and $\mathbb R$" --- Let $\{q_n: n\in\mathbb N\}=\mathbb Q$. -Consider the closed set -$L = \mathbb R\setminus \bigcup_{n\in \mathbb N} (q_n-2^{-n},q_n+2^{-n})$. -It is discrete, as a subset of the irrational numbers. -Since the Lebesgue measure of $L$ is infinite (its complement is of finite measure), $L$ is uncountable. - -As $L$ is also closed in Euclidean topology, any uncountable closed set in Euclidean topology is of cardinality $\mathfrak c$, by Cantor–Bendixson theorem. +Consider the set +$L = \mathbb R\setminus \bigcup_{n\in \mathbb N} (q_n-2^{-n},q_n+2^{-n})$, +which is closed in {S25} and hence closed in the finer topology of $X$. +It is discrete in $X$, as a subset of the irrational numbers. +And it is uncountable since its Lebesgue measure is infinite (its complement is of finite measure). +Since every uncountable closed set in {S25} has cardinality $\mathfrak c$ +(consequence of the Cantor-Bendixson theorem, see for example {{mathse:740095}}), +it follows that $|L|=\mathfrak c$. From 5d707a95a9b1b3057fb89f887a2a28ca7fccefd7 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Mon, 12 Jan 2026 18:28:26 -0500 Subject: [PATCH 08/10] fix build --- spaces/S000063/properties/P000227.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000063/properties/P000227.md b/spaces/S000063/properties/P000227.md index 0c96526e61..97e72a970a 100644 --- a/spaces/S000063/properties/P000227.md +++ b/spaces/S000063/properties/P000227.md @@ -4,7 +4,7 @@ property: P000227 value: true refs: - mathse: 740095 - name: "Bijection between closed uncountable subset of $\mathbb R$ and $\mathbb R$" + name: Bijection between closed uncountable subset of $\mathbb R$ and $\mathbb R$ --- Let $\{q_n: n\in\mathbb N\}=\mathbb Q$. From 065424b1203b06e3d9278ce80f6cb6abe5633850 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Tue, 13 Jan 2026 14:30:09 -0500 Subject: [PATCH 09/10] T833 update --- theorems/T000833.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/theorems/T000833.md b/theorems/T000833.md index 231b00363e..b7c56f8609 100644 --- a/theorems/T000833.md +++ b/theorems/T000833.md @@ -6,7 +6,8 @@ then: P000198: false --- -Since $\aleph_0 < \mathfrak{c} = 2^{\aleph_0}$. +If $X$ has a closed discrete subset of size continuum, +its extent satisfies $e(X)\ge\mathfrak c>\aleph_0$. ---- The converse is independent of ZFC. It is true iff CH holds. From 956f2768ffd3ae1f9bea6a328ad76f78523c36d2 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Tue, 13 Jan 2026 15:02:27 -0500 Subject: [PATCH 10/10] P227 subsets to sets --- properties/P000227.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/properties/P000227.md b/properties/P000227.md index 509233afe3..418aff8dc5 100644 --- a/properties/P000227.md +++ b/properties/P000227.md @@ -3,17 +3,17 @@ uid: P000227 name: Has a discrete closed subset of size $\mathfrak c$ --- -$X$ has a discrete closed subset of cardinality $\mathfrak c=2^{\aleph_0}$. +$X$ has a discrete closed set of cardinality $\mathfrak c=2^{\aleph_0}$. -*Note*: Discrete closed subsets are exactly the subsets with no limit point in $X$. +*Note*: The discrete closed sets are exactly the subsets with no limit point in $X$. *Note*: This property implies $e(X)\ge\mathfrak c$, -where the *extent* $e(X)$ is the supremum of the cardinality of discrete closed subsets in $X$. +where the *extent* $e(X)$ is the supremum of the cardinality of discrete closed sets in $X$. But, under certain set-theoretic assumptions (for example, if $\mathfrak c=\aleph_{\omega_1}$), -there are spaces with $e(X)=\mathfrak c$ and without discrete closed subset of cardinality $\mathfrak c$, +there are spaces with $e(X)=\mathfrak c$ and without discrete closed set of cardinality $\mathfrak c$, i.e., where the supremum is not attained. -Compare with these properties, where $D$ denotes a discrete closed subset in $X$: +Compare with these properties, where $D$ denotes a discrete closed set in $X$: - {P107} $(\exists D: |D|=1)$ - {P21} $(\forall D: |D|<\aleph_0)$ - {P198} $(\forall D: |D|\le\aleph_0)$