Add Q^ω, as a better distinguishment of Erdős space

spaces/S000030/properties/P000066.md

@@ -0,0 +1,8 @@
+---
+space: S000030
+property: P000066
+value: false
+---
+
+{S28} is homeomorphic to $\mathbb Z^\omega$, which can be viewed as a closed subspace of {S30}.
+It follows from {S28|P66} and the fact that the closed subspace of a {P66} space is {P66}.

spaces/S000142/README.md

@@ -14,6 +14,6 @@ The space $\ell^2$ is the Banach space of sequences of real numbers $x=(x_i)_i$
 equipped with the norm $\|x\|_2=(\Sigma_i x_i^2)^{1/2}$ and corresponding distance and topology.
 
 The space $\ell^2$ is topologically homeomorphic to {S30};
-but note the space $X$ is *not* homeomorphic to the subspace $\mathbb Q^\omega$ of $\mathbb R^\omega$.
+but note the space $X$ is *not* homeomorphic to the subspace {S146} of $\mathbb R^\omega$.
 
 See {{mathse:151954}} or Example 6.2.19 in {{zb:0684.54001}}.

spaces/S000146/README.md

@@ -0,0 +1,20 @@
+---
+uid: S000146
+name: Countable product of rationals $\mathbb Q^\omega$
+aliases:
+  - Q^ω
+  - Q^Z
+refs:
+  - mathse: 4416328
+    name: What spaces are homeomorphic to $\mathbb Q^\omega = \mathbb Q^\mathbb N = \mathbb Q^\infty$?
+  - zb: "0642.54033"
+    name: Characterizations of the countable infinite product of rationals and some related problems
+  - zb: "0562.54054"
+    name: Countable products of zero-dimensional absolute $F_{\sigma \delta}$ spaces
+---
+
+The countable product of copies of {S27}.
+
+Note that this space is not homeomorphic to {S142}.
+
+See {{mathse:4416328}}, {{zb:0642.54033}}, or {{zb:0562.54054}}.

spaces/S000146/properties/P000027.md

@@ -0,0 +1,7 @@
+---
+space: S000146
+property: P000027
+value: true
+---
+
+As a subspace of {S30}, it follows from that {S30|P27}.

spaces/S000146/properties/P000050.md

@@ -0,0 +1,9 @@
+---
+space: S000146
+property: P000050
+value: true
+---
+
+Since {S27|P50}, we can take a clopen basis of each point $x \in \mathbb Q$.
+
+Now for each point $\left( x_k \right)_{k \in \omega}$, we take $\prod_{k < \omega} U_k$ where $U_k = \mathbb Q$ for all $k$ but a finite $k$, which travels over the clopen basis of $x_k$.

spaces/S000146/properties/P000053.md

@@ -0,0 +1,7 @@
+---
+space: S000146
+property: P000053
+value: true
+---
+
+As a subspace of {S30}, it follows from that {S30|P53}.

spaces/S000146/properties/P000056.md

@@ -0,0 +1,7 @@
+---
+space: S000146
+property: P000056
+value: true
+---
+
+Let $F_q = \left\{ q \right\} \times \mathbb Q^\omega$, then $F_q$ is a closed set with empty interior, and $\mathbb Q^\omega = \bigcup_{q \in \mathbb Q} F_q$.

spaces/S000146/properties/P000065.md

@@ -0,0 +1,7 @@
+---
+space: S000146
+property: P000065
+value: true
+---
+
+$\left| \mathbb Q^\omega \right| = \mathfrak c$.

spaces/S000146/properties/P000066.md

@@ -0,0 +1,8 @@
+---
+space: S000146
+property: P000066
+value: false
+---
+
+{S28} is homeomorphic to $\mathbb Z^\omega$, which can be viewed as a closed subspace of {S146}.
+It follows from {S28|P66} and the fact that the closed subspace of a {P66} space is {P66}.

spaces/S000146/properties/P000087.md

@@ -0,0 +1,7 @@
+---
+space: S000146
+property: P000087
+value: true
+---
+
+This space is an additive subgroup of {S30}.

spaces/S000146/properties/P000133.md

@@ -0,0 +1,12 @@
+---
+space: S000146
+property: P000133
+value: true
+refs:
+  - zb: "0642.54033"
+    name: Characterizations of the countable infinite product of rationals and some related problems
+---
+
+In fact, {S146} can be viewed as a subspace of $\omega^\omega$ which $\lim_{n \to + \infty} x_n = \infty$, by {{zb:0642.54033}}.
+
+Then using the embedding of $\omega^\omega$ into $\mathbb R$ ({S28}), we know that {S146} can be viewed as a dense subspace of $\mathbb R$, which is {P133}.