add density leq continuum (#1016)

properties/P000209.md

@@ -0,0 +1,6 @@
+---
+uid: P000209
+name: Density $\leq\mathfrak c$
+---
+
+There exists a dense subset with cardinality $\leq \mathfrak c=2^{\aleph_0}=|\mathbb R|$.

spaces/S000174/properties/P000163.md

@@ -4,4 +4,4 @@ property: P000163
 value: true
 ---
 
-$|X| = \omega_1^\omega \leq (2^\omega)^\omega = 2^\omega = \mathfrak{c}$.
+$|X| = |\omega_1^\omega| \leq (2^\omega)^\omega = 2^\omega = \mathfrak{c}$.

spaces/S001103/properties/P000209.md

@@ -0,0 +1,16 @@
+---
+space: S001103
+property: P000209
+value: true
+refs:
+    - name: A remark on density characters (Hewitt)
+      doi: 10.1090/S0002-9904-1946-08613-9
+---
+
+Follows from {S2|P209}
+and {S2|P3},
+and the Hewitt-Marczewski-Pondiczery theorem
+({{doi:10.1090/S0002-9904-1946-08613-9}})
+(<https://planetmath.org/HewittMarczewskiPondiczeryTheorem>)
+which shows that {P209} is preserved by powers up to $2^{\mathfrak c}$
+for {P3} spaces.

theorems/T000404.md

@@ -2,9 +2,9 @@
 uid: T000404
 if:
   and:
-    - P000026: true
     - P000079: true
     - P000099: true
+    - P000209: true
 then:
   P000163: true
 refs:
@@ -16,4 +16,3 @@ refs:
 
 Proved in {{mathse:4850951}} with a generalization of the ideas in Theorem 4.4 of {{mr:0776620}}.
 
-Note also that density $\leq\mathfrak c$ is sufficient.

theorems/T000603.md

@@ -0,0 +1,9 @@
+---
+uid: T000603
+if:
+  P000026: true
+then:
+  P000209: true
+---
+
+Follows as every countable set has cardinality $\leq\mathfrak c$.

theorems/T000604.md

@@ -0,0 +1,9 @@
+---
+uid: T000604
+if:
+  P000163: true
+then:
+  P000209: true
+---
+
+Follows as every space is a dense subset of itself.