From 5dc5bff1153dad9fd689b4f6c4111c4d185bdb95 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Sun, 17 May 2026 16:57:40 -0400 Subject: [PATCH 1/3] S225: Closed half-plane --- spaces/S000225/README.md | 8 ++++++++ spaces/S000225/properties/P000016.md | 8 ++++++++ spaces/S000225/properties/P000089.md | 8 ++++++++ spaces/S000225/properties/P000120.md | 8 ++++++++ spaces/S000225/properties/P000123.md | 8 ++++++++ spaces/S000225/properties/P000184.md | 8 ++++++++ spaces/S000225/properties/P000199.md | 8 ++++++++ spaces/S000225/properties/P000204.md | 7 +++++++ spaces/S000225/properties/P000236.md | 8 ++++++++ 9 files changed, 71 insertions(+) create mode 100644 spaces/S000225/README.md create mode 100644 spaces/S000225/properties/P000016.md create mode 100644 spaces/S000225/properties/P000089.md create mode 100644 spaces/S000225/properties/P000120.md create mode 100644 spaces/S000225/properties/P000123.md create mode 100644 spaces/S000225/properties/P000184.md create mode 100644 spaces/S000225/properties/P000199.md create mode 100644 spaces/S000225/properties/P000204.md create mode 100644 spaces/S000225/properties/P000236.md diff --git a/spaces/S000225/README.md b/spaces/S000225/README.md new file mode 100644 index 000000000..9b07a75b6 --- /dev/null +++ b/spaces/S000225/README.md @@ -0,0 +1,8 @@ +--- +uid: S000225 +name: Closed half-plane $\mathbb R^2_+$ +--- + +The subspace $\mathbb R^2_+=\{(x,y)\in\mathbb R^2 : y\ge 0\}$ of {S176}. + +This space is homemorphic to the product of {S25} and {S210}. diff --git a/spaces/S000225/properties/P000016.md b/spaces/S000225/properties/P000016.md new file mode 100644 index 000000000..45b733c91 --- /dev/null +++ b/spaces/S000225/properties/P000016.md @@ -0,0 +1,8 @@ +--- +space: S000225 +property: P000016 +value: false +--- + + $X$ is not a bounded subset of $\mathbb R^2$. + \ No newline at end of file diff --git a/spaces/S000225/properties/P000089.md b/spaces/S000225/properties/P000089.md new file mode 100644 index 000000000..93340fd1f --- /dev/null +++ b/spaces/S000225/properties/P000089.md @@ -0,0 +1,8 @@ +--- +space: S000225 +property: P000089 +value: false +--- + + The map $(x,y)\mapsto (x+1,y)$ has no fixed point. + \ No newline at end of file diff --git a/spaces/S000225/properties/P000120.md b/spaces/S000225/properties/P000120.md new file mode 100644 index 000000000..254fdb4bf --- /dev/null +++ b/spaces/S000225/properties/P000120.md @@ -0,0 +1,8 @@ +--- +space: S000225 +property: P000120 +value: false +--- + +$X$ contains an open subspace homeomorphic to {S176} +and {S176|P120}. diff --git a/spaces/S000225/properties/P000123.md b/spaces/S000225/properties/P000123.md new file mode 100644 index 000000000..cc66e082f --- /dev/null +++ b/spaces/S000225/properties/P000123.md @@ -0,0 +1,8 @@ +--- +space: S000225 +property: P000123 +value: false +--- + + The point $(0,0)\in X$ does not have a neighborhood homeomorphic to some $\mathbb R^n$. + \ No newline at end of file diff --git a/spaces/S000225/properties/P000184.md b/spaces/S000225/properties/P000184.md new file mode 100644 index 000000000..4c692af2b --- /dev/null +++ b/spaces/S000225/properties/P000184.md @@ -0,0 +1,8 @@ +--- +space: S000225 +property: P000184 +value: true +--- + + By definition. + \ No newline at end of file diff --git a/spaces/S000225/properties/P000199.md b/spaces/S000225/properties/P000199.md new file mode 100644 index 000000000..9307680ee --- /dev/null +++ b/spaces/S000225/properties/P000199.md @@ -0,0 +1,8 @@ +--- +space: S000225 +property: P000199 +value: true +--- + + $X$ is a convex subset of $\mathbb R^2$. + \ No newline at end of file diff --git a/spaces/S000225/properties/P000204.md b/spaces/S000225/properties/P000204.md new file mode 100644 index 000000000..0c74547fb --- /dev/null +++ b/spaces/S000225/properties/P000204.md @@ -0,0 +1,7 @@ +--- +space: S000225 +property: P000204 +value: false +--- + +By inspection. diff --git a/spaces/S000225/properties/P000236.md b/spaces/S000225/properties/P000236.md new file mode 100644 index 000000000..9cb77f98f --- /dev/null +++ b/spaces/S000225/properties/P000236.md @@ -0,0 +1,8 @@ +--- +space: S000225 +property: P000236 +value: true +--- + + $X$ satisfies {P236} with $n=2$. + \ No newline at end of file From 5d33927e8cd269046abb97c2a3b945ad4e185eb2 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Sun, 17 May 2026 17:14:32 -0400 Subject: [PATCH 2/3] P123 -> P122 --- spaces/S000225/properties/P000016.md | 5 ++--- spaces/S000225/properties/P000089.md | 5 ++--- spaces/S000225/properties/P000122.md | 7 +++++++ spaces/S000225/properties/P000123.md | 8 -------- spaces/S000225/properties/P000184.md | 5 ++--- spaces/S000225/properties/P000199.md | 5 ++--- spaces/S000225/properties/P000236.md | 5 ++--- 7 files changed, 17 insertions(+), 23 deletions(-) create mode 100644 spaces/S000225/properties/P000122.md delete mode 100644 spaces/S000225/properties/P000123.md diff --git a/spaces/S000225/properties/P000016.md b/spaces/S000225/properties/P000016.md index 45b733c91..1e3f38631 100644 --- a/spaces/S000225/properties/P000016.md +++ b/spaces/S000225/properties/P000016.md @@ -3,6 +3,5 @@ space: S000225 property: P000016 value: false --- - - $X$ is not a bounded subset of $\mathbb R^2$. - \ No newline at end of file + +$X$ is not a bounded subset of $\mathbb R^2$. diff --git a/spaces/S000225/properties/P000089.md b/spaces/S000225/properties/P000089.md index 93340fd1f..4128e501b 100644 --- a/spaces/S000225/properties/P000089.md +++ b/spaces/S000225/properties/P000089.md @@ -3,6 +3,5 @@ space: S000225 property: P000089 value: false --- - - The map $(x,y)\mapsto (x+1,y)$ has no fixed point. - \ No newline at end of file + +The map $(x,y)\mapsto (x+1,y)$ has no fixed point. diff --git a/spaces/S000225/properties/P000122.md b/spaces/S000225/properties/P000122.md new file mode 100644 index 000000000..cce3f02dd --- /dev/null +++ b/spaces/S000225/properties/P000122.md @@ -0,0 +1,7 @@ +--- +space: S000225 +property: P000122 +value: false +--- + +The point $(0,0)\in X$ does not have a neighborhood homeomorphic to some $\mathbb R^n$. diff --git a/spaces/S000225/properties/P000123.md b/spaces/S000225/properties/P000123.md deleted file mode 100644 index cc66e082f..000000000 --- a/spaces/S000225/properties/P000123.md +++ /dev/null @@ -1,8 +0,0 @@ ---- -space: S000225 -property: P000123 -value: false ---- - - The point $(0,0)\in X$ does not have a neighborhood homeomorphic to some $\mathbb R^n$. - \ No newline at end of file diff --git a/spaces/S000225/properties/P000184.md b/spaces/S000225/properties/P000184.md index 4c692af2b..07083bba0 100644 --- a/spaces/S000225/properties/P000184.md +++ b/spaces/S000225/properties/P000184.md @@ -3,6 +3,5 @@ space: S000225 property: P000184 value: true --- - - By definition. - \ No newline at end of file + +By definition. diff --git a/spaces/S000225/properties/P000199.md b/spaces/S000225/properties/P000199.md index 9307680ee..13a1a806d 100644 --- a/spaces/S000225/properties/P000199.md +++ b/spaces/S000225/properties/P000199.md @@ -3,6 +3,5 @@ space: S000225 property: P000199 value: true --- - - $X$ is a convex subset of $\mathbb R^2$. - \ No newline at end of file + +$X$ is a convex subset of $\mathbb R^2$. diff --git a/spaces/S000225/properties/P000236.md b/spaces/S000225/properties/P000236.md index 9cb77f98f..b8b90a7ce 100644 --- a/spaces/S000225/properties/P000236.md +++ b/spaces/S000225/properties/P000236.md @@ -3,6 +3,5 @@ space: S000225 property: P000236 value: true --- - - $X$ satisfies {P236} with $n=2$. - \ No newline at end of file + +$X$ satisfies {P236} with $n=2$. From 9c551f9d3722fe51a14996a3c435939e725b0b28 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Sun, 17 May 2026 23:15:51 -0400 Subject: [PATCH 3/3] changes from review --- spaces/S000225/README.md | 2 +- spaces/S000225/properties/P000199.md | 3 ++- spaces/S000225/properties/P000204.md | 7 ++++++- 3 files changed, 9 insertions(+), 3 deletions(-) diff --git a/spaces/S000225/README.md b/spaces/S000225/README.md index 9b07a75b6..fb38a9154 100644 --- a/spaces/S000225/README.md +++ b/spaces/S000225/README.md @@ -1,6 +1,6 @@ --- uid: S000225 -name: Closed half-plane $\mathbb R^2_+$ +name: Closed upper half-plane $\mathbb R^2_+$ --- The subspace $\mathbb R^2_+=\{(x,y)\in\mathbb R^2 : y\ge 0\}$ of {S176}. diff --git a/spaces/S000225/properties/P000199.md b/spaces/S000225/properties/P000199.md index 13a1a806d..b34fa4f0d 100644 --- a/spaces/S000225/properties/P000199.md +++ b/spaces/S000225/properties/P000199.md @@ -4,4 +4,5 @@ property: P000199 value: true --- -$X$ is a convex subset of $\mathbb R^2$. +Since $X$ is a convex subset of $\mathbb R^2$, +it can be deformation retracted to a point using a straight-line homotopy. diff --git a/spaces/S000225/properties/P000204.md b/spaces/S000225/properties/P000204.md index 0c74547fb..49aa7e955 100644 --- a/spaces/S000225/properties/P000204.md +++ b/spaces/S000225/properties/P000204.md @@ -2,6 +2,11 @@ space: S000225 property: P000204 value: false +refs: + - mathse: 214317 + name: $(X\times Y)\setminus (A\times B)$ is connected for $A,B$ proper subsets of connected $X,Y$ --- -By inspection. +$X$ is the product of two connected spaces, each with at least two points. +Such a product minus a point is always connected +(see {{mathse:214317}}).