From b8eed8d2e4f3211f503543f2d19854cbab12f683 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 27 Dec 2024 00:59:05 -0500 Subject: [PATCH 1/3] Introduce S209 Circle with two origins --- spaces/S000209/README.md | 10 ++++++++++ spaces/S000209/properties/P000016.md | 7 +++++++ spaces/S000209/properties/P000038.md | 7 +++++++ spaces/S000209/properties/P000101.md | 7 +++++++ spaces/S000209/properties/P000123.md | 7 +++++++ spaces/S000209/properties/P000169.md | 7 +++++++ spaces/S000209/properties/P000200.md | 10 ++++++++++ spaces/S000209/properties/P000204.md | 7 +++++++ 8 files changed, 62 insertions(+) create mode 100644 spaces/S000209/README.md create mode 100644 spaces/S000209/properties/P000016.md create mode 100644 spaces/S000209/properties/P000038.md create mode 100644 spaces/S000209/properties/P000101.md create mode 100644 spaces/S000209/properties/P000123.md create mode 100644 spaces/S000209/properties/P000169.md create mode 100644 spaces/S000209/properties/P000200.md create mode 100644 spaces/S000209/properties/P000204.md diff --git a/spaces/S000209/README.md b/spaces/S000209/README.md new file mode 100644 index 000000000..522149328 --- /dev/null +++ b/spaces/S000209/README.md @@ -0,0 +1,10 @@ +--- +uid: S000209 +name: Circle with two origins +--- + +Choose a point $0 \in S^1$ to be called the origin, and replace $0$ with two origins $0_1$ and $0_2$. Basic open neighborhoods of a point $x \neq 0$ are Euclidean open neighborhoods of $x$ not containing $0$. Basic open neighborhoods of each origin $0_i$ are of the form $(U\setminus\{0\})\cup\{0_i\}$ with $U$ a Euclidean open neighborhood of $0$. + +Let $\{1, 2\}$ have the discrete topology. $X$ is homeomorphic to the quotient space of $S^1 \times \{1, 2\}$ obtained by identifying $\langle \theta, 1 \rangle$ and $\langle \theta, 2 \rangle$ exactly when $\theta \not\equiv 0 \mod 2\pi$. + +$X$ is homeomorphic to the one-point compactification of {S83}. diff --git a/spaces/S000209/properties/P000016.md b/spaces/S000209/properties/P000016.md new file mode 100644 index 000000000..aaadd8588 --- /dev/null +++ b/spaces/S000209/properties/P000016.md @@ -0,0 +1,7 @@ +--- +space: S000209 +property: P000016 +value: true +--- + +$X$ is homeomorphic to the one-point compactification of {S83}. diff --git a/spaces/S000209/properties/P000038.md b/spaces/S000209/properties/P000038.md new file mode 100644 index 000000000..1955cf66a --- /dev/null +++ b/spaces/S000209/properties/P000038.md @@ -0,0 +1,7 @@ +--- +space: S000209 +property: P000038 +value: true +--- + +The map $[0, 2\pi] \to X$ defined by $t \mapsto \langle t, 1 \rangle$ if $t < 2\pi$ and $2\pi \mapsto \langle 2\pi, 2\rangle$ is an injective and continuous. It is clear by restricting this to sub-intervals and reparameterizing that every pair of points is connected by an injective path. diff --git a/spaces/S000209/properties/P000101.md b/spaces/S000209/properties/P000101.md new file mode 100644 index 000000000..36845731c --- /dev/null +++ b/spaces/S000209/properties/P000101.md @@ -0,0 +1,7 @@ +--- +space: S000209 +property: P000101 +value: false +--- + +Same argument as {S83|P101}. diff --git a/spaces/S000209/properties/P000123.md b/spaces/S000209/properties/P000123.md new file mode 100644 index 000000000..5601ba5e4 --- /dev/null +++ b/spaces/S000209/properties/P000123.md @@ -0,0 +1,7 @@ +--- +space: S000209 +property: P000123 +value: true +--- + +Each point is contained in an open set homeomorphic to $S^1$, namely $X\setminus\{0_1\}$ or $X\setminus\{0_2\}$, and {S170|P123}. diff --git a/spaces/S000209/properties/P000169.md b/spaces/S000209/properties/P000169.md new file mode 100644 index 000000000..2ecd45073 --- /dev/null +++ b/spaces/S000209/properties/P000169.md @@ -0,0 +1,7 @@ +--- +space: S000209 +property: P000169 +value: false +--- + +Same argument as {S83|P169}. diff --git a/spaces/S000209/properties/P000200.md b/spaces/S000209/properties/P000200.md new file mode 100644 index 000000000..9dfba3f04 --- /dev/null +++ b/spaces/S000209/properties/P000200.md @@ -0,0 +1,10 @@ +--- +space: S000209 +property: P000200 +value: false +refs: + - zb: "1044.55001" + name: Algebraic Topology (Hatcher) +--- + +The map sending the origins to $0_1$ and fixing all other points is a retraction onto {S170}. Since {S170|P200}, it follows by Proposition 1.17 of {{zb:1044.55001}} that $X$ is not simply connected. diff --git a/spaces/S000209/properties/P000204.md b/spaces/S000209/properties/P000204.md new file mode 100644 index 000000000..376580d59 --- /dev/null +++ b/spaces/S000209/properties/P000204.md @@ -0,0 +1,7 @@ +--- +space: S000209 +property: P000204 +value: false +--- + +For any $p \in X$, $X \backslash \{p\}$ is either homeomorphic to {S83} or {S170}. From acee523949d2b507808f731c8fcf040109aa56a4 Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 27 Dec 2024 01:20:42 -0500 Subject: [PATCH 2/3] Typo / minor writing improvement --- spaces/S000209/properties/P000038.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000209/properties/P000038.md b/spaces/S000209/properties/P000038.md index 1955cf66a..631a504b9 100644 --- a/spaces/S000209/properties/P000038.md +++ b/spaces/S000209/properties/P000038.md @@ -4,4 +4,4 @@ property: P000038 value: true --- -The map $[0, 2\pi] \to X$ defined by $t \mapsto \langle t, 1 \rangle$ if $t < 2\pi$ and $2\pi \mapsto \langle 2\pi, 2\rangle$ is an injective and continuous. It is clear by restricting this to sub-intervals and reparameterizing that every pair of points is connected by an injective path. +The map $[0, 2\pi] \to X$ defined by $t \mapsto \langle t, 1 \rangle$ if $t < 2\pi$ and $2\pi \mapsto \langle 2\pi, 2\rangle$ is injective and continuous. It is clear by restricting this map to sub-intervals and reparameterizing the results that every pair of points is connected by an injective path. From 4bdd1f5a9b82e90660fed5c99cb10303776e5ebe Mon Sep 17 00:00:00 2001 From: Geoffrey Sangston Date: Fri, 27 Dec 2024 08:33:25 -0500 Subject: [PATCH 3/3] Apply Moniker's suggestions 1 --- spaces/S000209/README.md | 7 +++++-- spaces/S000209/properties/P000016.md | 5 ++++- 2 files changed, 9 insertions(+), 3 deletions(-) diff --git a/spaces/S000209/README.md b/spaces/S000209/README.md index 522149328..6c2f11619 100644 --- a/spaces/S000209/README.md +++ b/spaces/S000209/README.md @@ -1,10 +1,13 @@ --- uid: S000209 name: Circle with two origins +refs: + - wikipedia: Alexandroff_extension + name: Alexandroff extension on Wikipedia --- Choose a point $0 \in S^1$ to be called the origin, and replace $0$ with two origins $0_1$ and $0_2$. Basic open neighborhoods of a point $x \neq 0$ are Euclidean open neighborhoods of $x$ not containing $0$. Basic open neighborhoods of each origin $0_i$ are of the form $(U\setminus\{0\})\cup\{0_i\}$ with $U$ a Euclidean open neighborhood of $0$. -Let $\{1, 2\}$ have the discrete topology. $X$ is homeomorphic to the quotient space of $S^1 \times \{1, 2\}$ obtained by identifying $\langle \theta, 1 \rangle$ and $\langle \theta, 2 \rangle$ exactly when $\theta \not\equiv 0 \mod 2\pi$. +Let $\{1, 2\}$ have the discrete topology. $X$ is homeomorphic to the quotient space of $S^1 \times \{1, 2\}$ obtained by identifying $\langle \theta, 1 \rangle$ and $\langle \theta, 2 \rangle$ exactly when $\theta {\not\equiv} 0 \mod 2\pi$. -$X$ is homeomorphic to the one-point compactification of {S83}. +$X$ is homeomorphic to the Alexandroff extension of {S83}. diff --git a/spaces/S000209/properties/P000016.md b/spaces/S000209/properties/P000016.md index aaadd8588..08ced2745 100644 --- a/spaces/S000209/properties/P000016.md +++ b/spaces/S000209/properties/P000016.md @@ -2,6 +2,9 @@ space: S000209 property: P000016 value: true +refs: + - wikipedia: Alexandroff_extension + name: Alexandroff extension on Wikipedia --- -$X$ is homeomorphic to the one-point compactification of {S83}. +$X$ is homeomorphic to the Alexandroff extension of {S83}.