Has closed discrete subset of size 𝔠 (part 2)

properties/P000227.md

@@ -1,18 +1,24 @@
 ---
 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 $\mathfrak c=2^{\aleph_0}$.
 
-*Note*: Discrete closed sets are exactly the sets 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 sets in $X$.
-But there are spaces with $e(X)=\mathfrak c$ and without discrete closed set of cardinality $\mathfrak c$,
+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 set of cardinality $\mathfrak c$,
 i.e., where the supremum is not attained.
 
 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.

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 a closed and discrete subspace of cardinality $\mathfrak c$.

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.

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 $\mathfrak c$.

spaces/S000063/properties/P000227.md

@@ -0,0 +1,18 @@
+---
+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 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 $|L|=\mathfrak c$.

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 $\mathfrak c$.

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 $\mathfrak c$.

spaces/S000070/properties/P000227.md

@@ -0,0 +1,7 @@
+---
+space: S000070
+property: P000227
+value: true
+---
+
+The subspace $L$ is closed and discrete with cardinality $\mathfrak c$.

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 $\mathfrak c$.

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 $\mathfrak c$.

spaces/S000085/properties/P000198.md → 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 $2^\mathfrak c$.

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 $\mathfrak c$.

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 $\mathfrak c$.

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 $\mathfrak c$.

spaces/S000110/properties/P000198.md → 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., $|M| = 2^\mathfrak{c}$.

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 $\mathfrak c$.

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 $\mathfrak c$.

spaces/S000134/properties/P000198.md → 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 $\epsilon\le a$.

spaces/S000135/properties/P000198.md → spaces/S000135/properties/P000227.md

@@ -1,12 +1,12 @@
 ---
 space: S000135
-property: P000198
-value: false
+property: P000227
+value: true
 refs:
 - zb: "0386.54001"
   name: Counterexamples in Topology
 ---
 
 This topology is finer than {S134},
 (cf. item #1 for space #141 in {{zb:0386.54001}})
-and {S134|P198}.
+and {S134|P227}.

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 $\mathfrak c$.

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 $\mathfrak c$.

spaces/S000214/properties/P000198.md → spaces/S000214/properties/P000227.md

@@ -1,7 +1,7 @@
 ---
 space: S000214
-property: P000198
-value: false
+property: P000227
+value: true
 ---
 
 The subset $\{f\in X: f(\lambda)=0\text{ for }\lambda\in[\omega,\omega_1)\}$

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 $\mathfrak{c}$.

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.