From 2ce33893dd8c3e4a89800131aa5eb1ed6a982fa2 Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Wed, 6 Nov 2024 15:18:00 +0800 Subject: [PATCH 1/4] Merge S106 into S30, resolve #829 --- spaces/S000030/README.md | 8 ++++++++ spaces/S000106/README.md | 15 --------------- spaces/S000106/properties/P000017.md | 10 ---------- spaces/S000106/properties/P000023.md | 10 ---------- spaces/S000106/properties/P000026.md | 10 ---------- spaces/S000106/properties/P000038.md | 10 ---------- spaces/S000106/properties/P000043.md | 10 ---------- spaces/S000106/properties/P000053.md | 13 ------------- spaces/S000106/properties/P000055.md | 11 ----------- spaces/S000106/properties/P000087.md | 10 ---------- spaces/S000106/properties/P000129.md | 7 ------- 11 files changed, 8 insertions(+), 106 deletions(-) delete mode 100644 spaces/S000106/README.md delete mode 100644 spaces/S000106/properties/P000017.md delete mode 100644 spaces/S000106/properties/P000023.md delete mode 100644 spaces/S000106/properties/P000026.md delete mode 100644 spaces/S000106/properties/P000038.md delete mode 100644 spaces/S000106/properties/P000043.md delete mode 100644 spaces/S000106/properties/P000053.md delete mode 100644 spaces/S000106/properties/P000055.md delete mode 100644 spaces/S000106/properties/P000087.md delete mode 100644 spaces/S000106/properties/P000129.md diff --git a/spaces/S000030/README.md b/spaces/S000030/README.md index bfcfa59f28..15b18b2c8f 100644 --- a/spaces/S000030/README.md +++ b/spaces/S000030/README.md @@ -3,7 +3,11 @@ uid: S000030 name: Hilbert space aliases: - $\ell^2$ + - $\mathbb R^\omega$ - Fréchet space + - Continuous real-valued functions on the unit interval + - C(I) + - C[0,1] counterexamples_id: 36 refs: - doi: 10.1007/978-1-4612-6290-9 @@ -19,3 +23,7 @@ This space is homeomorphic to $\mathbb{R}^\omega$ with the product topology (Fr Defined as counterexamples #36 ("Hilbert Space") and #37 ("Fréchet Space") in {{doi:10.1007/978-1-4612-6290-9}}. + +--- +It follows from {{wikipedia:Anderson–Kadec theorem}} that any infinite-dimensional separable Banach/Fréchet space is homeomorphic to $\mathbb R^\omega$. +So $\mathbb R^\omega$, $\ell^p$ ($1 \leq p < + \infty$) and $C[0, 1]$ are all homeomorphic in the view of topological spaces. diff --git a/spaces/S000106/README.md b/spaces/S000106/README.md deleted file mode 100644 index 869a768661..0000000000 --- a/spaces/S000106/README.md +++ /dev/null @@ -1,15 +0,0 @@ ---- -uid: S000106 -name: Continuous real-valued functions on the unit interval -aliases: - - C(I) - - C[0,1] -counterexamples_id: 108 -refs: - - doi: 10.1007/978-1-4612-6290-9 - name: Counterexamples in Topology ---- -The space of continuous real-valued functions \(C(I)\) or \(C(I,\mathbb R)\). Its topology is generated by the sup norm \(d(f,g) = \sup|f(t) - g(t)|\). - -Defined as counterexample #108 ("\(C[0,1]\)") -in {{doi:10.1007/978-1-4612-6290-9}}. diff --git a/spaces/S000106/properties/P000017.md b/spaces/S000106/properties/P000017.md deleted file mode 100644 index 27f8c62103..0000000000 --- a/spaces/S000106/properties/P000017.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000106 -property: P000017 -value: false -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -See item #3 for space #108 in {{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000106/properties/P000023.md b/spaces/S000106/properties/P000023.md deleted file mode 100644 index a117665dfc..0000000000 --- a/spaces/S000106/properties/P000023.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000106 -property: P000023 -value: false -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -See item #3 for space #108 in {{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000106/properties/P000026.md b/spaces/S000106/properties/P000026.md deleted file mode 100644 index f7ba7186ea..0000000000 --- a/spaces/S000106/properties/P000026.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000106 -property: P000026 -value: true -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -See item #2 for space #108 in {{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000106/properties/P000038.md b/spaces/S000106/properties/P000038.md deleted file mode 100644 index 8c92c75e18..0000000000 --- a/spaces/S000106/properties/P000038.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000106 -property: P000038 -value: true -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -See item #4 for space #108 in {{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000106/properties/P000043.md b/spaces/S000106/properties/P000043.md deleted file mode 100644 index e56b9a2c7d..0000000000 --- a/spaces/S000106/properties/P000043.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000106 -property: P000043 -value: true -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -See item #4 for space #108 in {{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000106/properties/P000053.md b/spaces/S000106/properties/P000053.md deleted file mode 100644 index 3c78880355..0000000000 --- a/spaces/S000106/properties/P000053.md +++ /dev/null @@ -1,13 +0,0 @@ ---- -space: S000106 -property: P000053 -value: true -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -The topology is defined by the sup norm $d(f,g)=sup|f(t)−g(t)|$. - -Asserted in the General Reference Chart for space #108 in -{{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000106/properties/P000055.md b/spaces/S000106/properties/P000055.md deleted file mode 100644 index d98ae65203..0000000000 --- a/spaces/S000106/properties/P000055.md +++ /dev/null @@ -1,11 +0,0 @@ ---- -space: S000106 -property: P000055 -value: true -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -Asserted in the General Reference Chart for space #108 in -{{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000106/properties/P000087.md b/spaces/S000106/properties/P000087.md deleted file mode 100644 index 50b1ac23eb..0000000000 --- a/spaces/S000106/properties/P000087.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000106 -property: P000087 -value: true -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -The space is a normed linear space. In particular, it is a topological group with addition. diff --git a/spaces/S000106/properties/P000129.md b/spaces/S000106/properties/P000129.md deleted file mode 100644 index ade61becd9..0000000000 --- a/spaces/S000106/properties/P000129.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000106 -property: P000129 -value: false ---- - -The space is non-trivial by definition. From 43a4f2355cb1d5c5221b285341446cabc083a07e Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Sat, 9 Nov 2024 11:16:49 +0800 Subject: [PATCH 2/4] Update README.md Add original references from *Counterexamples in Topology* for S106 --- spaces/S000030/README.md | 6 ++---- 1 file changed, 2 insertions(+), 4 deletions(-) diff --git a/spaces/S000030/README.md b/spaces/S000030/README.md index 15b18b2c8f..6c538fbc2b 100644 --- a/spaces/S000030/README.md +++ b/spaces/S000030/README.md @@ -21,9 +21,7 @@ Let $X$ be the space $\ell^2$ of all sequences $x = \langle x_i \rangle$ of real 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}}. -Defined as counterexamples #36 ("Hilbert Space") and #37 ("Fréchet Space") -in {{doi:10.1007/978-1-4612-6290-9}}. - ---- It follows from {{wikipedia:Anderson–Kadec theorem}} that any infinite-dimensional separable Banach/Fréchet space is homeomorphic to $\mathbb R^\omega$. So $\mathbb R^\omega$, $\ell^p$ ($1 \leq p < + \infty$) and $C[0, 1]$ are all homeomorphic in the view of topological spaces. + +Defined as counterexamples #36 ("Hilbert Space") and #37 ("Fréchet Space") in {{doi:10.1007/978-1-4612-6290-9}}. Also defined as counterexample #108 ("\(C[0,1]\)") in {{doi:10.1007/978-1-4612-6290-9}}. From 035c846d3679b69e28d1550b18245bdc49d9b5e9 Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Sun, 10 Nov 2024 08:36:25 +0800 Subject: [PATCH 3/4] Apply suggestions from code review Co-authored-by: Steven Clontz --- spaces/S000030/README.md | 13 +++++-------- 1 file changed, 5 insertions(+), 8 deletions(-) diff --git a/spaces/S000030/README.md b/spaces/S000030/README.md index 6c538fbc2b..79dd3192cb 100644 --- a/spaces/S000030/README.md +++ b/spaces/S000030/README.md @@ -1,13 +1,10 @@ --- uid: S000030 -name: Hilbert space +name: Hilbert space $\ell^2$ aliases: - - $\ell^2$ - - $\mathbb R^\omega$ - - Fréchet space + - Fréchet space $\mathbb R^\omega$ - Continuous real-valued functions on the unit interval - - C(I) - - C[0,1] + - $C([0,1])$ counterexamples_id: 36 refs: - doi: 10.1007/978-1-4612-6290-9 @@ -21,7 +18,7 @@ Let $X$ be the space $\ell^2$ of all sequences $x = \langle x_i \rangle$ of real 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}}. -It follows from {{wikipedia:Anderson–Kadec theorem}} that any infinite-dimensional separable Banach/Fréchet space is homeomorphic to $\mathbb R^\omega$. +It follows from {{wikipedia:Anderson–Kadec_theorem}} that any infinite-dimensional separable Banach/Fréchet space is homeomorphic to $\mathbb R^\omega$. So $\mathbb R^\omega$, $\ell^p$ ($1 \leq p < + \infty$) and $C[0, 1]$ are all homeomorphic in the view of topological spaces. -Defined as counterexamples #36 ("Hilbert Space") and #37 ("Fréchet Space") in {{doi:10.1007/978-1-4612-6290-9}}. Also defined as counterexample #108 ("\(C[0,1]\)") in {{doi:10.1007/978-1-4612-6290-9}}. +Defined as counterexamples #36 ("Hilbert Space"), #37 ("Fréchet Space"), and #108 ("\(C[0,1]\)") in {{doi:10.1007/978-1-4612-6290-9}}. From 658267a408b2794fb406ab06fee70a5a20ed9e94 Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Sun, 10 Nov 2024 10:55:28 +0800 Subject: [PATCH 4/4] Update spaces/S000030/README.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- spaces/S000030/README.md | 11 ++++++++--- 1 file changed, 8 insertions(+), 3 deletions(-) diff --git a/spaces/S000030/README.md b/spaces/S000030/README.md index 79dd3192cb..d818ce8327 100644 --- a/spaces/S000030/README.md +++ b/spaces/S000030/README.md @@ -18,7 +18,12 @@ Let $X$ be the space $\ell^2$ of all sequences $x = \langle x_i \rangle$ of real 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}}. -It follows from {{wikipedia:Anderson–Kadec_theorem}} that any infinite-dimensional separable Banach/Fréchet space is homeomorphic to $\mathbb R^\omega$. -So $\mathbb R^\omega$, $\ell^p$ ($1 \leq p < + \infty$) and $C[0, 1]$ are all homeomorphic in the view of topological spaces. +More generally, any infinite-dimensional separable Banach space or separable Fréchet space is homeomorphic to $\mathbb R^\omega$. See {{wikipedia:Anderson–Kadec theorem}}. -Defined as counterexamples #36 ("Hilbert Space"), #37 ("Fréchet Space"), and #108 ("\(C[0,1]\)") in {{doi:10.1007/978-1-4612-6290-9}}. +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 + +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}}.