Has closed discrete subset of size 𝔠 (part 4)

properties/P000198.md

@@ -16,3 +16,8 @@ That is, every closed discrete subspace of $X$ is countable.
 Equivalently, every uncountable set $S\subseteq X$ has a limit point in $X$.
 
 See {{mr:0776620}} or section a-3 of {{doi:10.1016/B978-0-444-50355-8.X5000-4}}.
+
+----
+#### Meta-properties
+
+- If $X$ is covered by countably many subspaces, each having the property, then so does $X$.

properties/P000227.md

@@ -22,3 +22,4 @@ Compare with these properties, where $D$ denotes a discrete closed set in $X$:
 #### Meta-properties
 
 - This property is preserved in any finer topology.
+- If a closed subspace of $X$ satisfies this property, so does $X$.

spaces/S000077/properties/P000227.md

@@ -0,0 +1,7 @@
+---
+space: S000077
+property: P000227
+value: true
+---
+
+$X$ contains {S63} as a closed subspace, and {S63|P227}.

spaces/S000079/properties/P000198.md

@@ -4,11 +4,8 @@ property: P000198
 value: true
 ---
 
-$X$ is a countable union of subspaces
-$[0,\omega_1]\times\{n\}$ for $n<\omega$ and $[0,\omega_1)\times\{\omega\}$ which are homeomorphic to {S36}
-and {S35}, respectively.
-Every closed and discrete subset of $X$ has
-countable intersection with each of them
-({S36|P198}
- and {S35|P198})
-and therefore is countable itself.
+$X$ is covered by the countably many subspaces
+$[0,\omega_1]\times\{n\}$ for $n<\omega$ and $[0,\omega_1)\times\{\omega\}$,
+with each of them {P198}
+since {S36|P198}
+and {S35|P198}.

spaces/S000081/properties/P000198.md

@@ -0,0 +1,14 @@
+---
+space: S000081
+property: P000198
+value: true
+refs:
+- mathse: 3652735
+  name: If $X$ is Lindelof and $Y$ is compact, then $X \times Y$ is Lindelof
+---
+
+$X$ is the union of finitely many subspaces, each with {P198}:
+* $[0, \omega_1] \times [-1, 0)$, because it is {P18} as the product of a {P16} space and a {P18} space (see for example {{mathse:3652735}}), and {T562}.
+* $[0, \omega_1] \times (0, 1]$ for the same reason.
+* $[0, \omega_1) \times \{0\}$, because it is homeomorphic to {S35} and {S35|P198}.
+* $\{\left< \omega_1, 0 \right>\}$, because {S162|P198}.

spaces/S000101/properties/P000227.md

@@ -0,0 +1,12 @@
+---
+space: S000101
+property: P000227
+value: true
+refs:
+  - zb: "0684.54001"
+    name: General Topology (Engelking, 1989)
+---
+
+See Exercise 3.1.H(a) in {{zb:0684.54001}}.
+
+See also <https://dantopology.wordpress.com/2014/03/08/looking-for-a-closed-and-discrete-subspace-of-a-product-space/>.

spaces/S000107/properties/P000227.md

@@ -0,0 +1,7 @@
+---
+space: S000107
+property: P000227
+value: true
+---
+
+$\left\{ 0, 1 \right\}^\omega$ is a closed and discrete subset of cardinality $\mathfrak c$.

spaces/S000136/properties/P000227.md

@@ -0,0 +1,15 @@
+---
+space: S000136
+property: P000227
+value: true
+refs:
+- zb: "0386.54001"
+  name: Counterexamples in Topology
+---
+
+$X$ contains {S137} as a closed subspace, and {S137|P227}.
+
+Explicitly, $M$ is a closed and discrete subset of cardinality $\mathfrak c$.
+
+See item #1 for space #142 in {{zb:0386.54001}}.
+

spaces/S000137/properties/P000227.md

@@ -0,0 +1,12 @@
+---
+space: S000137
+property: P000227
+value: true
+refs:
+- zb: "0386.54001"
+  name: Counterexamples in Topology
+---
+
+$M$ is a closed and discrete subset of cardinality $\mathfrak c$.
+
+See item #1 for space #143 in {{zb:0386.54001}}.

spaces/S000153/properties/P000198.md

@@ -0,0 +1,9 @@
+---
+space: S000153
+property: P000198
+value: true
+---
+
+$X$ is the union of the subspaces $\{0\} \times (0, 1)$ and $\left[ 1, \omega_1 \right) \times [0, 1)$,
+which are homeomorphic to {S25} and {S38}, respectively.
+Each has countable extent, as {S25|P198} and {S38|P198}.

spaces/S001103/properties/P000227.md

@@ -0,0 +1,7 @@
+---
+space: S001103
+property: P000227
+value: true
+---
+
+$X$ contains a closed subspace homeomorphic to {S101}, and {S101|P227}.