Adding spaces homeomorphic to S30 to its description

spaces/S000030/README.md

@@ -3,8 +3,9 @@ uid: S000030
 name: Hilbert space $\ell^2$
 aliases:
   - Fréchet space $\mathbb R^\omega$
-  - Continuous real-valued functions on the unit interval
   - $C([0,1])$
+  - Unit sphere of $\ell^2$
+  - Closed unit ball of $\ell^2$
 counterexamples_id: 36
 refs:
   - doi: 10.1007/978-1-4612-6290-9 
@@ -13,17 +14,27 @@ refs:
     name: Hilbert space on Wikipedia
   - doi: 10.1090/S0002-9904-1968-12044-0
     name: A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines (Anderson & Bing)
+  - mr: 0478168
+    name: Selected topics in infinite-dimensional topology (Bessaga & Pełczyński)
 ---
 Let $X$ be the space $\ell^2$ of all sequences $x = \langle x_i \rangle$ of real numbers such that $\Sigma x_i^2$ converges, with topology generated by the metric $d(x,y) = \sqrt{\Sigma(x_i - y_i)^2}$.  Any separable Hilbert space is isomorphic to $\ell^2$.
 
 This space is homeomorphic to $\mathbb{R}^\omega$ with the product topology (Fréchet space).  For this deep result due to R. D. Anderson, see for example {{doi:10.1090/S0002-9904-1968-12044-0}}.
 
-More generally, any infinite-dimensional separable Banach space or separable Fréchet space is homeomorphic to $\mathbb R^\omega$.  See {{wikipedia:Anderson–Kadec theorem}}.
+More generally, any infinite-dimensional separable Banach space or, even more generally, any infinite-dimensional separable Fréchet space is homeomorphic to $\mathbb R^\omega$. This is stated in Theorem VI.5.2 (Anderson-Kadec) on page 189 of {{mr:0478168}}. See also {{wikipedia:Anderson–Kadec theorem}}.
 
 In particular, the following are all homeomorphic as topological spaces:
 - $\mathbb R^\omega$
 - Sequence spaces $\ell^p$ ($1 \leq p < + \infty$)
 - Spaces $L^p([0,1])$ ($1 \leq p < + \infty$) of Lebesgue measurable functions with finite $L^p$-norm
-- the Banach space $C([0,1])$ of real-valued continuous functions on $[0,1]$ with the supremum norm
+- The Banach space $C([0,1])$ of real-valued continuous functions on $[0,1]$ with the supremum norm
+
+Moreover, by Theorem VI.2.1, Corollary VI.2.2, and Theorem VI.5.1 of {{mr:0478168}} and Corollary 9.3 of {{doi:10.1090/S0002-9904-1968-12044-0}}, the following are also homeomorphic to $\mathbb R^\omega$:
+- The closed upper half-space of $\ell^2$, i.e., $\{\langle x_i \rangle \in \ell^2: x_1 \geq 0\}$, given the subspace topology
+- The subspace of $\ell^2$ consisting of all elements with infinitely many nonzero entries, i.e., $\{\langle x_i \rangle \in \ell^2: x_i \neq 0 \text{ for infinitely many } i\}$, given the subspace topology
+- The cone over $\mathbb R^\omega$, i.e., the quotient of $\mathbb R^\omega \times [0, 1]$ with $\mathbb R^\omega \times \{0\}$ identified to a single point
+- The open cone over $\mathbb R^\omega$, i.e., the quotient of $\mathbb R^\omega \times [0, \infty)$ with $\mathbb R^\omega \times \{0\}$ identified to a single point
+- The unit sphere of any infinite-dimensional separable Banach space, i.e., $\{x \in E: \|x\| = 1\}$ where $E$ is an infinite-dimensional separable Banach space under the norm $\|\cdot\|$, given the topology generated by the metric $d(x,y) = \|x-y\|$; in particular, the unit sphere of $\ell^2$
+- The closed unit ball of any infinite-dimensional separable Banach space, i.e., $\{x \in E: \|x\| \leq 1\}$ where $E$ is an infinite-dimensional separable Banach space under the norm $\|\cdot\|$, given the topology generated by the metric $d(x,y) = \|x-y\|$; in particular, the closed unit ball of $\ell^2$
 
 Specific instances of the above are counterexamples #36 ("Hilbert Space"), #37 ("Fréchet Space"), and #108 ("$C[0,1]$")  in {{doi:10.1007/978-1-4612-6290-9}}.