From b2c9f573b7a83b4a1c30b09ec3e8514ff139c2a4 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Thu, 16 Apr 2026 11:10:11 -0400 Subject: [PATCH 01/11] Definitions --- spaces/S000106/README.md | 32 ++++++++++++++++++++++++++++++++ 1 file changed, 32 insertions(+) create mode 100644 spaces/S000106/README.md diff --git a/spaces/S000106/README.md b/spaces/S000106/README.md new file mode 100644 index 000000000..163bc3797 --- /dev/null +++ b/spaces/S000106/README.md @@ -0,0 +1,32 @@ +--- +uid: S000106 +name: Direct limit $\mathbb R^\infty$ of Euclidean spaces $\mathbb R^n$ +refs: + - wikipedia: Direct_limit + name: Direct limit on Wikipedia + - mathse: 3961052 + name: Answer to "Is the weak topology on $\mathbb{R}^{\infty}$ the same as the box topology?" + - mathse: 5012784 + name: Answer to "Is $\ell^\infty$ with box topology connected?" +--- +The subset $\mathbb{R}^\infty$ of eventually $0$ sequences in $\mathbb{R}^\omega$, with the finest +topology such that the standard inclusion maps $\mathbb{R}^n \hookrightarrow \mathbb{R}^\infty$, +$x \mapsto (x^1, \ldots, x^n, 0, \ldots)$, are continuous for each $n$, where $\mathbb{R}^n$ has +the Euclidean topology. + +Equivalently, the set $U \subset \mathbb{R}^\infty$ is open if and only if $U \cap \mathbb{R}^n$ +is open in $\mathbb{R}^n$ for each $n$, where we identify each Euclidean space $\mathbb{R}^n$ with +its image. + +Equivalently, $\mathbb{R}^\infty$ is the direct limit $\varinjlim \mathbb{R}^n$ of the directed +system consisting of Euclidean spaces and standard inclusion maps +$\mathbb{R}^i \hookrightarrow \mathbb{R}^j$, $x \mapsto (x^1, \ldots, x^i, 0, \ldots)$, +for each $i < j$. + +Equivalently, $\mathbb{R}^\infty \subset \mathbb{R}^\omega$ has the subspace topology, where +$\mathbb{R}^\omega$ is given the box topology; this is shown in {{mathse:3961052}}. Moreover, +it is shown in {{mathse:5012784}} that $\mathbb{R}^\infty$ is a quasi-component of the origin in +$\mathbb{R}^\omega$. Hence $\mathbb{R}^\infty$ embeds into {{S107}} +as a path component. + +For general discussion on direct limits, see {{wikipedia:Direct_limit}}. From cf55d410c7214abd2d6ff1bd89fefa114047b14b Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Thu, 16 Apr 2026 11:53:58 -0400 Subject: [PATCH 02/11] Includes reference to the definition --- spaces/S000106/README.md | 12 ++++++++++++ 1 file changed, 12 insertions(+) diff --git a/spaces/S000106/README.md b/spaces/S000106/README.md index 163bc3797..6686170f1 100644 --- a/spaces/S000106/README.md +++ b/spaces/S000106/README.md @@ -8,6 +8,14 @@ refs: name: Answer to "Is the weak topology on $\mathbb{R}^{\infty}$ the same as the box topology?" - mathse: 5012784 name: Answer to "Is $\ell^\infty$ with box topology connected?" + - zb: "0298.57008" + name: Characteristic classes (Milnor-Stasheff) + - zb: "0307.55015" + name: Fibre bundles. 2nd ed. (Husemoller) + - zb: "1280.54001" + name: Geometric aspects of general topology. (Sakai) + - wikipedia: Fréchet–Urysohn_space + name: Fréchet–Urysohn space on Wikipedia --- The subset $\mathbb{R}^\infty$ of eventually $0$ sequences in $\mathbb{R}^\omega$, with the finest topology such that the standard inclusion maps $\mathbb{R}^n \hookrightarrow \mathbb{R}^\infty$, @@ -30,3 +38,7 @@ $\mathbb{R}^\omega$. Hence $\mathbb{R}^\infty$ embeds into {{S107}} as a path component. For general discussion on direct limits, see {{wikipedia:Direct_limit}}. + +Defined on page 62 of {{zb:"0298.57008"}}, on page 2 of {{zb:"0307.55015"}}, +on page 56 of {{zb:"1280.54001}}, and on {{wikipedia:Fréchet–Urysohn_space}} +under Direct limit of finite-dimensional Euclidean spaces. From 23b1c41f551c8423c873f5cda192ebc6989e4860 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 1 May 2026 00:31:52 -0400 Subject: [PATCH 03/11] real TVS + CW structure --- spaces/S000106/properties/P000238.md | 16 ++++++++++++++++ spaces/S000106/properties/P000240.md | 11 +++++++++++ 2 files changed, 27 insertions(+) create mode 100644 spaces/S000106/properties/P000238.md create mode 100644 spaces/S000106/properties/P000240.md diff --git a/spaces/S000106/properties/P000238.md b/spaces/S000106/properties/P000238.md new file mode 100644 index 000000000..bb7e15a01 --- /dev/null +++ b/spaces/S000106/properties/P000238.md @@ -0,0 +1,16 @@ +--- +space: S000106 +property: P000238 +value: true +refs: + - zb: "1280.54001" + name: Geometric aspects of general topology. (Sakai) +--- + +$\mathbb{R}^\infty$ with the natural scalar multiplication and addition operations is a vector space +over $\mathbb{R}$. It remains to argue each operation is continuous. + +For each $n$, the scalar multiplication function $\mathbb{R} \times \mathbb{R}^\infty \to \mathbb{R}^\infty$ restricted to $\mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n \hookrightarrow \mathbb{R}^\infty$ is continuous, since the first map identifies with the usual scalar multiplication of $n$-dimensional Euclidean space. Since [S25|P130], Proposition 2.8.3 of {{zb:"1280.54001"}} implies $\mathbb{R} \times \mathbb{R}^\infty \cong \varinjlim (\mathbb{R} \times \mathbb{R}^n)$. This means that $\mathbb{R} \times \mathbb{R}^\infty$ has the final topology with respect to the inclusions +$\mathbb{R} \times \mathbb{R}^n \hookrightarrow \mathbb{R} \times \mathbb{R}^\infty$. Thus $\mathbb{R} \times \mathbb{R}^\infty \to \mathbb{R}^\infty$ is continuous. + +Since each $\mathbb{R}^n$ is locally compact, Proposition 2.8.4 of {{zb:"1280.54001"}} implies $\mathbb{R}^\infty \times \mathbb{R}^\infty \cong \varinjlim (\mathbb{R}^n \times \mathbb{R}^n)$. Making a similar argument as above shows that the natural addition operation on $\mathbb{R}^\infty$ is continuous. \ No newline at end of file diff --git a/spaces/S000106/properties/P000240.md b/spaces/S000106/properties/P000240.md new file mode 100644 index 000000000..d8c5fa640 --- /dev/null +++ b/spaces/S000106/properties/P000240.md @@ -0,0 +1,11 @@ +--- +space: S000106 +property: P000240 +value: true +refs: + - zb: "1280.54001" + name: Geometric aspects of general topology. (Sakai) +--- + +The chain of subspaces $\empty \subset \mathbb{R}^0 \subset \mathbb{R}^1 \cdots$ is a CW structure. + From d408eccbb5149a7e4ae526c854e47889c090d054 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 1 May 2026 10:29:05 -0400 Subject: [PATCH 04/11] Adjustments --- spaces/S000106/README.md | 16 +++++++--------- spaces/S000106/properties/P000238.md | 2 +- 2 files changed, 8 insertions(+), 10 deletions(-) diff --git a/spaces/S000106/README.md b/spaces/S000106/README.md index 6686170f1..1c7ed8541 100644 --- a/spaces/S000106/README.md +++ b/spaces/S000106/README.md @@ -17,19 +17,19 @@ refs: - wikipedia: Fréchet–Urysohn_space name: Fréchet–Urysohn space on Wikipedia --- -The subset $\mathbb{R}^\infty$ of eventually $0$ sequences in $\mathbb{R}^\omega$, with the finest -topology such that the standard inclusion maps $\mathbb{R}^n \hookrightarrow \mathbb{R}^\infty$, -$x \mapsto (x^1, \ldots, x^n, 0, \ldots)$, are continuous for each $n$, where $\mathbb{R}^n$ has -the Euclidean topology. +$X$ is the subset $\mathbb{R}^\infty$ of eventually $0$ sequences in $\mathbb{R}^\omega$, with the final +topology with respect to the standard inclusion maps $\mathbb{R}^n \hookrightarrow \mathbb{R}^\infty$, +$x \mapsto (x^1, \ldots, x^n, 0, \ldots)$. By definition of final topology, this means that $\mathbb{R}^\infty$ has the finest topology such that each such inclusion map is continuous. Equivalently, the set $U \subset \mathbb{R}^\infty$ is open if and only if $U \cap \mathbb{R}^n$ is open in $\mathbb{R}^n$ for each $n$, where we identify each Euclidean space $\mathbb{R}^n$ with -its image. +its image. -Equivalently, $\mathbb{R}^\infty$ is the direct limit $\varinjlim \mathbb{R}^n$ of the directed +Equivalently, $\mathbb{R}^\infty$ is the direct limit $\varinjlim \mathbb{R}^n := (\bigsqcup_{i = 1}^\infty \mathbb{R}^i)/\sim$ of the directed system consisting of Euclidean spaces and standard inclusion maps $\mathbb{R}^i \hookrightarrow \mathbb{R}^j$, $x \mapsto (x^1, \ldots, x^i, 0, \ldots)$, -for each $i < j$. +for each $i < j$. The equivalence relation $\sim$ identifies points which correspond under these inclusions. +For general discussion on direct limits, see {{wikipedia:Direct_limit}}. Equivalently, $\mathbb{R}^\infty \subset \mathbb{R}^\omega$ has the subspace topology, where $\mathbb{R}^\omega$ is given the box topology; this is shown in {{mathse:3961052}}. Moreover, @@ -37,8 +37,6 @@ it is shown in {{mathse:5012784}} that $\mathbb{R}^\infty$ is a quasi-component $\mathbb{R}^\omega$. Hence $\mathbb{R}^\infty$ embeds into {{S107}} as a path component. -For general discussion on direct limits, see {{wikipedia:Direct_limit}}. - Defined on page 62 of {{zb:"0298.57008"}}, on page 2 of {{zb:"0307.55015"}}, on page 56 of {{zb:"1280.54001}}, and on {{wikipedia:Fréchet–Urysohn_space}} under Direct limit of finite-dimensional Euclidean spaces. diff --git a/spaces/S000106/properties/P000238.md b/spaces/S000106/properties/P000238.md index bb7e15a01..e8365640d 100644 --- a/spaces/S000106/properties/P000238.md +++ b/spaces/S000106/properties/P000238.md @@ -11,6 +11,6 @@ $\mathbb{R}^\infty$ with the natural scalar multiplication and addition operatio over $\mathbb{R}$. It remains to argue each operation is continuous. For each $n$, the scalar multiplication function $\mathbb{R} \times \mathbb{R}^\infty \to \mathbb{R}^\infty$ restricted to $\mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n \hookrightarrow \mathbb{R}^\infty$ is continuous, since the first map identifies with the usual scalar multiplication of $n$-dimensional Euclidean space. Since [S25|P130], Proposition 2.8.3 of {{zb:"1280.54001"}} implies $\mathbb{R} \times \mathbb{R}^\infty \cong \varinjlim (\mathbb{R} \times \mathbb{R}^n)$. This means that $\mathbb{R} \times \mathbb{R}^\infty$ has the final topology with respect to the inclusions -$\mathbb{R} \times \mathbb{R}^n \hookrightarrow \mathbb{R} \times \mathbb{R}^\infty$. Thus $\mathbb{R} \times \mathbb{R}^\infty \to \mathbb{R}^\infty$ is continuous. +$\mathbb{R} \times \mathbb{R}^n \hookrightarrow \mathbb{R} \times \mathbb{R}^\infty$. By the universal property of the final topology, it follows that $\mathbb{R} \times \mathbb{R}^\infty \to \mathbb{R}^\infty$ is continuous. Since each $\mathbb{R}^n$ is locally compact, Proposition 2.8.4 of {{zb:"1280.54001"}} implies $\mathbb{R}^\infty \times \mathbb{R}^\infty \cong \varinjlim (\mathbb{R}^n \times \mathbb{R}^n)$. Making a similar argument as above shows that the natural addition operation on $\mathbb{R}^\infty$ is continuous. \ No newline at end of file From 71aa4dd1b682dcd59d0671cc5908eefe351b3f74 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Sat, 2 May 2026 11:23:40 -0400 Subject: [PATCH 05/11] Update spaces/S000106/properties/P000238.md Co-authored-by: yhx-12243 --- spaces/S000106/properties/P000238.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000106/properties/P000238.md b/spaces/S000106/properties/P000238.md index e8365640d..b92050704 100644 --- a/spaces/S000106/properties/P000238.md +++ b/spaces/S000106/properties/P000238.md @@ -10,7 +10,7 @@ refs: $\mathbb{R}^\infty$ with the natural scalar multiplication and addition operations is a vector space over $\mathbb{R}$. It remains to argue each operation is continuous. -For each $n$, the scalar multiplication function $\mathbb{R} \times \mathbb{R}^\infty \to \mathbb{R}^\infty$ restricted to $\mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n \hookrightarrow \mathbb{R}^\infty$ is continuous, since the first map identifies with the usual scalar multiplication of $n$-dimensional Euclidean space. Since [S25|P130], Proposition 2.8.3 of {{zb:"1280.54001"}} implies $\mathbb{R} \times \mathbb{R}^\infty \cong \varinjlim (\mathbb{R} \times \mathbb{R}^n)$. This means that $\mathbb{R} \times \mathbb{R}^\infty$ has the final topology with respect to the inclusions +For each $n$, the scalar multiplication function $\mathbb{R} \times \mathbb{R}^\infty \to \mathbb{R}^\infty$ restricted to $\mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n \hookrightarrow \mathbb{R}^\infty$ is continuous, since the first map identifies with the usual scalar multiplication of $n$-dimensional Euclidean space. Since {S25|P130}, Proposition 2.8.3 of {{zb:1280.54001}} implies $\mathbb{R} \times \mathbb{R}^\infty \cong \varinjlim (\mathbb{R} \times \mathbb{R}^n)$. This means that $\mathbb{R} \times \mathbb{R}^\infty$ has the final topology with respect to the inclusions $\mathbb{R} \times \mathbb{R}^n \hookrightarrow \mathbb{R} \times \mathbb{R}^\infty$. By the universal property of the final topology, it follows that $\mathbb{R} \times \mathbb{R}^\infty \to \mathbb{R}^\infty$ is continuous. Since each $\mathbb{R}^n$ is locally compact, Proposition 2.8.4 of {{zb:"1280.54001"}} implies $\mathbb{R}^\infty \times \mathbb{R}^\infty \cong \varinjlim (\mathbb{R}^n \times \mathbb{R}^n)$. Making a similar argument as above shows that the natural addition operation on $\mathbb{R}^\infty$ is continuous. \ No newline at end of file From 885138fa4d0a4456dee739a3f7f3587fb16a312c Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Sat, 2 May 2026 11:23:49 -0400 Subject: [PATCH 06/11] Update spaces/S000106/properties/P000238.md Co-authored-by: yhx-12243 --- spaces/S000106/properties/P000238.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000106/properties/P000238.md b/spaces/S000106/properties/P000238.md index b92050704..395bacf5f 100644 --- a/spaces/S000106/properties/P000238.md +++ b/spaces/S000106/properties/P000238.md @@ -13,4 +13,4 @@ over $\mathbb{R}$. It remains to argue each operation is continuous. For each $n$, the scalar multiplication function $\mathbb{R} \times \mathbb{R}^\infty \to \mathbb{R}^\infty$ restricted to $\mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n \hookrightarrow \mathbb{R}^\infty$ is continuous, since the first map identifies with the usual scalar multiplication of $n$-dimensional Euclidean space. Since {S25|P130}, Proposition 2.8.3 of {{zb:1280.54001}} implies $\mathbb{R} \times \mathbb{R}^\infty \cong \varinjlim (\mathbb{R} \times \mathbb{R}^n)$. This means that $\mathbb{R} \times \mathbb{R}^\infty$ has the final topology with respect to the inclusions $\mathbb{R} \times \mathbb{R}^n \hookrightarrow \mathbb{R} \times \mathbb{R}^\infty$. By the universal property of the final topology, it follows that $\mathbb{R} \times \mathbb{R}^\infty \to \mathbb{R}^\infty$ is continuous. -Since each $\mathbb{R}^n$ is locally compact, Proposition 2.8.4 of {{zb:"1280.54001"}} implies $\mathbb{R}^\infty \times \mathbb{R}^\infty \cong \varinjlim (\mathbb{R}^n \times \mathbb{R}^n)$. Making a similar argument as above shows that the natural addition operation on $\mathbb{R}^\infty$ is continuous. \ No newline at end of file +Since each $\mathbb{R}^n$ is locally compact, Proposition 2.8.4 of {{zb:1280.54001}} implies $\mathbb{R}^\infty \times \mathbb{R}^\infty \cong \varinjlim (\mathbb{R}^n \times \mathbb{R}^n)$. Making a similar argument as above shows that the natural addition operation on $\mathbb{R}^\infty$ is continuous. \ No newline at end of file From 9996bcb26341c6a8d204467fce67b401f71d1577 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Sat, 2 May 2026 11:24:02 -0400 Subject: [PATCH 07/11] Update spaces/S000106/README.md Co-authored-by: yhx-12243 --- spaces/S000106/README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/spaces/S000106/README.md b/spaces/S000106/README.md index 1c7ed8541..c71c644b2 100644 --- a/spaces/S000106/README.md +++ b/spaces/S000106/README.md @@ -37,6 +37,6 @@ it is shown in {{mathse:5012784}} that $\mathbb{R}^\infty$ is a quasi-component $\mathbb{R}^\omega$. Hence $\mathbb{R}^\infty$ embeds into {{S107}} as a path component. -Defined on page 62 of {{zb:"0298.57008"}}, on page 2 of {{zb:"0307.55015"}}, -on page 56 of {{zb:"1280.54001}}, and on {{wikipedia:Fréchet–Urysohn_space}} +Defined on page 62 of {{zb:0298.57008}}, on page 2 of {{zb:0307.55015}}, +on page 56 of {{zb:1280.54001}}, and on {{wikipedia:Fréchet–Urysohn_space}} under Direct limit of finite-dimensional Euclidean spaces. From 139cae33d7b4fcd9550534b0a9c8f8831a896128 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Sat, 2 May 2026 11:24:36 -0400 Subject: [PATCH 08/11] Update spaces/S000106/README.md Co-authored-by: yhx-12243 --- spaces/S000106/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000106/README.md b/spaces/S000106/README.md index c71c644b2..8bb7f7d75 100644 --- a/spaces/S000106/README.md +++ b/spaces/S000106/README.md @@ -34,7 +34,7 @@ For general discussion on direct limits, see {{wikipedia:Direct_limit}}. Equivalently, $\mathbb{R}^\infty \subset \mathbb{R}^\omega$ has the subspace topology, where $\mathbb{R}^\omega$ is given the box topology; this is shown in {{mathse:3961052}}. Moreover, it is shown in {{mathse:5012784}} that $\mathbb{R}^\infty$ is a quasi-component of the origin in -$\mathbb{R}^\omega$. Hence $\mathbb{R}^\infty$ embeds into {{S107}} +$\mathbb{R}^\omega$. Hence $\mathbb{R}^\infty$ embeds into {S107} as a path component. Defined on page 62 of {{zb:0298.57008}}, on page 2 of {{zb:0307.55015}}, From e06247e8059ce9a4825e65ab08f5df63e084e968 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Sun, 3 May 2026 09:41:23 -0400 Subject: [PATCH 09/11] Update spaces/S000106/README.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- spaces/S000106/README.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/spaces/S000106/README.md b/spaces/S000106/README.md index 8bb7f7d75..c933e0607 100644 --- a/spaces/S000106/README.md +++ b/spaces/S000106/README.md @@ -17,13 +17,13 @@ refs: - wikipedia: Fréchet–Urysohn_space name: Fréchet–Urysohn space on Wikipedia --- -$X$ is the subset $\mathbb{R}^\infty$ of eventually $0$ sequences in $\mathbb{R}^\omega$, with the final -topology with respect to the standard inclusion maps $\mathbb{R}^n \hookrightarrow \mathbb{R}^\infty$, -$x \mapsto (x^1, \ldots, x^n, 0, \ldots)$. By definition of final topology, this means that $\mathbb{R}^\infty$ has the finest topology such that each such inclusion map is continuous. - -Equivalently, the set $U \subset \mathbb{R}^\infty$ is open if and only if $U \cap \mathbb{R}^n$ -is open in $\mathbb{R}^n$ for each $n$, where we identify each Euclidean space $\mathbb{R}^n$ with -its image. +$X$ is the subset $\mathbb{R}^\infty$ of eventually $0$ sequences in $\mathbb{R}^\omega$, +with the [final topology](https://en.wikipedia.org/wiki/Final_topology) +with respect to the standard inclusion maps $\mathbb{R}^n \hookrightarrow \mathbb{R}^\infty$, +$x \mapsto (x^1, \ldots, x^n, 0, \ldots)$. +Thus, a set $U \subseteq \mathbb{R}^\infty$ is open iff $U \cap \mathbb{R}^n$ +is open in $\mathbb{R}^n$ for each $n$, +where we identify each Euclidean space $\mathbb{R}^n$ with its image. Equivalently, $\mathbb{R}^\infty$ is the direct limit $\varinjlim \mathbb{R}^n := (\bigsqcup_{i = 1}^\infty \mathbb{R}^i)/\sim$ of the directed system consisting of Euclidean spaces and standard inclusion maps From 809431e9fe7756e3d25955f0e1111b8212185d83 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Sun, 3 May 2026 18:51:02 -0400 Subject: [PATCH 10/11] Update spaces/S000106/README.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- spaces/S000106/README.md | 9 ++++----- 1 file changed, 4 insertions(+), 5 deletions(-) diff --git a/spaces/S000106/README.md b/spaces/S000106/README.md index c933e0607..53fe22b22 100644 --- a/spaces/S000106/README.md +++ b/spaces/S000106/README.md @@ -25,11 +25,10 @@ Thus, a set $U \subseteq \mathbb{R}^\infty$ is open iff $U \cap \mathbb{R}^n$ is open in $\mathbb{R}^n$ for each $n$, where we identify each Euclidean space $\mathbb{R}^n$ with its image. -Equivalently, $\mathbb{R}^\infty$ is the direct limit $\varinjlim \mathbb{R}^n := (\bigsqcup_{i = 1}^\infty \mathbb{R}^i)/\sim$ of the directed -system consisting of Euclidean spaces and standard inclusion maps -$\mathbb{R}^i \hookrightarrow \mathbb{R}^j$, $x \mapsto (x^1, \ldots, x^i, 0, \ldots)$, -for each $i < j$. The equivalence relation $\sim$ identifies points which correspond under these inclusions. -For general discussion on direct limits, see {{wikipedia:Direct_limit}}. +The space $\mathbb{R}^\infty$ is the [direct limit](https://en.wikipedia.org/wiki/Direct_limit) +$\varinjlim \mathbb{R}^n$ of the direct system consisting of Euclidean spaces $\mathbb R^n$ +and standard inclusion maps $\mathbb{R}^n \hookrightarrow \mathbb{R}^m$, +$x \mapsto (x^1, \ldots, x^n, 0, \ldots,0)$, for each $n < m$. Equivalently, $\mathbb{R}^\infty \subset \mathbb{R}^\omega$ has the subspace topology, where $\mathbb{R}^\omega$ is given the box topology; this is shown in {{mathse:3961052}}. Moreover, From 09becd89c3d6401d3d335dc4e2a3f4393110161d Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Thu, 7 May 2026 02:07:02 -0400 Subject: [PATCH 11/11] add notation for box product S107 --- spaces/S000107/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000107/README.md b/spaces/S000107/README.md index 3eb8224cf..0cec522b3 100644 --- a/spaces/S000107/README.md +++ b/spaces/S000107/README.md @@ -1,6 +1,6 @@ --- uid: S000107 -name: Countable box product of reals +name: Countable box product of reals $\square^\omega\mathbb R$ aliases: - Countable boolean product of reals - Box product topology on R^ω