From fb49f2342dcc71f54ca3c3d37f49dc7409320b4a Mon Sep 17 00:00:00 2001 From: Batixx Date: Fri, 26 Dec 2025 16:11:30 +0100 Subject: [PATCH 1/7] completing --- properties/P000068.md | 1 + spaces/S000036/properties/P000068.md | 7 +++++++ spaces/S000141/properties/P000068.md | 7 +++++++ spaces/S000141/properties/P000103.md | 7 +++++++ 4 files changed, 22 insertions(+) create mode 100644 spaces/S000036/properties/P000068.md create mode 100644 spaces/S000141/properties/P000068.md create mode 100644 spaces/S000141/properties/P000103.md diff --git a/properties/P000068.md b/properties/P000068.md index eb094a9b92..f7eacbddc6 100644 --- a/properties/P000068.md +++ b/properties/P000068.md @@ -18,3 +18,4 @@ See section 6 of {{doi:10.1016/0166-8641(95)00067-4}}. #### Meta-properties - This property is hereditary with respect to closed subspaces. +- This property is hereditary with respect to cofinite subspaces (If $Y\subseteq X$ is cofinite and Rothberger, use $\mathcal{U}_1, \dots, \mathcal{U}_{|X\setminus Y|}$ to cover $X\setminus Y$ and then cover $Y$). diff --git a/spaces/S000036/properties/P000068.md b/spaces/S000036/properties/P000068.md new file mode 100644 index 0000000000..827f361846 --- /dev/null +++ b/spaces/S000036/properties/P000068.md @@ -0,0 +1,7 @@ +--- +space: S000036 +property: P000068 +value: false +--- + +{S35} is cofinite in $X$ and {S35|P68}. diff --git a/spaces/S000141/properties/P000068.md b/spaces/S000141/properties/P000068.md new file mode 100644 index 0000000000..db6a5b4d71 --- /dev/null +++ b/spaces/S000141/properties/P000068.md @@ -0,0 +1,7 @@ +--- +space: S000141 +property: P000068 +value: false +--- + +{S36} is a closed subspace of $X$ and {S36|P68}. diff --git a/spaces/S000141/properties/P000103.md b/spaces/S000141/properties/P000103.md new file mode 100644 index 0000000000..2ba533c8b5 --- /dev/null +++ b/spaces/S000141/properties/P000103.md @@ -0,0 +1,7 @@ +--- +space: S000141 +property: P000103 +value: false +--- + +$X$ contains {S36}, {S36|P19}, but the subspace is not closed in $X$. From 33a1ae7f89140cb6ac1c1d3aae88a6d53edc93ab Mon Sep 17 00:00:00 2001 From: Batixx Date: Fri, 26 Dec 2025 16:14:02 +0100 Subject: [PATCH 2/7] remove redundant --- spaces/S000141/properties/P000020.md | 7 ------- spaces/S000141/properties/P000081.md | 7 ------- spaces/S000141/properties/P000126.md | 8 -------- 3 files changed, 22 deletions(-) delete mode 100644 spaces/S000141/properties/P000020.md delete mode 100644 spaces/S000141/properties/P000081.md delete mode 100644 spaces/S000141/properties/P000126.md diff --git a/spaces/S000141/properties/P000020.md b/spaces/S000141/properties/P000020.md deleted file mode 100644 index 31f133c40d..0000000000 --- a/spaces/S000141/properties/P000020.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000141 -property: P000020 -value: true ---- - -$X$ the union of its two {P20} subspaces $\omega_1$ and $1+\omega^*$: {S35|P20} and {S20|P20}. diff --git a/spaces/S000141/properties/P000081.md b/spaces/S000141/properties/P000081.md deleted file mode 100644 index 7daaefb561..0000000000 --- a/spaces/S000141/properties/P000081.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000141 -property: P000081 -value: false ---- - -{P81} is a hereditary property. The space contains the subspace $\omega_1+1$, and {S36|P81}. diff --git a/spaces/S000141/properties/P000126.md b/spaces/S000141/properties/P000126.md deleted file mode 100644 index 1f097f48b1..0000000000 --- a/spaces/S000141/properties/P000126.md +++ /dev/null @@ -1,8 +0,0 @@ ---- -space: S000141 -property: P000126 -value: false ---- - -This space contains {S35} as a subspace, and {S35|P126}. -Therefore {S141} is not {P126}. From a21d2cf71f6246b223ea6f12907d3b194e93903c Mon Sep 17 00:00:00 2001 From: Batixx Date: Fri, 26 Dec 2025 17:27:59 +0100 Subject: [PATCH 3/7] fix --- properties/P000068.md | 1 - spaces/S000036/properties/P000068.md | 7 ------- spaces/S000036/properties/P000151.md | 7 +++++++ spaces/S000141/properties/P000068.md | 7 ------- spaces/S000141/properties/P000151.md | 7 +++++++ 5 files changed, 14 insertions(+), 15 deletions(-) delete mode 100644 spaces/S000036/properties/P000068.md create mode 100644 spaces/S000036/properties/P000151.md delete mode 100644 spaces/S000141/properties/P000068.md create mode 100644 spaces/S000141/properties/P000151.md diff --git a/properties/P000068.md b/properties/P000068.md index f7eacbddc6..eb094a9b92 100644 --- a/properties/P000068.md +++ b/properties/P000068.md @@ -18,4 +18,3 @@ See section 6 of {{doi:10.1016/0166-8641(95)00067-4}}. #### Meta-properties - This property is hereditary with respect to closed subspaces. -- This property is hereditary with respect to cofinite subspaces (If $Y\subseteq X$ is cofinite and Rothberger, use $\mathcal{U}_1, \dots, \mathcal{U}_{|X\setminus Y|}$ to cover $X\setminus Y$ and then cover $Y$). diff --git a/spaces/S000036/properties/P000068.md b/spaces/S000036/properties/P000068.md deleted file mode 100644 index 827f361846..0000000000 --- a/spaces/S000036/properties/P000068.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000036 -property: P000068 -value: false ---- - -{S35} is cofinite in $X$ and {S35|P68}. diff --git a/spaces/S000036/properties/P000151.md b/spaces/S000036/properties/P000151.md new file mode 100644 index 0000000000..e33b558bdb --- /dev/null +++ b/spaces/S000036/properties/P000151.md @@ -0,0 +1,7 @@ +--- +space: S000036 +property: P000151 +value: false +--- + +Player I can first pick the point $\omega_1 \in X$, then Player II has to pick a cocountauble neighborhood of $\omega_1 \in X$, giving Player I a winning strategy by {T341} and {T354}. diff --git a/spaces/S000141/properties/P000068.md b/spaces/S000141/properties/P000068.md deleted file mode 100644 index db6a5b4d71..0000000000 --- a/spaces/S000141/properties/P000068.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000141 -property: P000068 -value: false ---- - -{S36} is a closed subspace of $X$ and {S36|P68}. diff --git a/spaces/S000141/properties/P000151.md b/spaces/S000141/properties/P000151.md new file mode 100644 index 0000000000..dbb1a9a5a6 --- /dev/null +++ b/spaces/S000141/properties/P000151.md @@ -0,0 +1,7 @@ +--- +space: S000141 +property: P000151 +value: true +--- + +Similar to {S36 |P151}. From 9b4588b655fd99e4094c968ddc5fb443ee5147c4 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Sat, 27 Dec 2025 02:20:37 +0100 Subject: [PATCH 4/7] Update spaces/S000141/properties/P000151.md Co-authored-by: yhx-12243 --- spaces/S000141/properties/P000151.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000141/properties/P000151.md b/spaces/S000141/properties/P000151.md index dbb1a9a5a6..4f118c0d84 100644 --- a/spaces/S000141/properties/P000151.md +++ b/spaces/S000141/properties/P000151.md @@ -4,4 +4,4 @@ property: P000151 value: true --- -Similar to {S36 |P151}. +Similar to {S36|P151}. From c179d820bbf267d2e052994abb483ae643c6b092 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Sat, 27 Dec 2025 08:04:55 +0100 Subject: [PATCH 5/7] Update spaces/S000141/properties/P000103.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- spaces/S000141/properties/P000103.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000141/properties/P000103.md b/spaces/S000141/properties/P000103.md index 2ba533c8b5..941f51b49a 100644 --- a/spaces/S000141/properties/P000103.md +++ b/spaces/S000141/properties/P000103.md @@ -4,4 +4,4 @@ property: P000103 value: false --- -$X$ contains {S36}, {S36|P19}, but the subspace is not closed in $X$. +$X$ contains {S36} as a subspace, and {S36|P103}. From 07414d85dbd48e8bb12bb24ef8f916283aeadbc4 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sat, 27 Dec 2025 10:10:54 +0100 Subject: [PATCH 6/7] hopefully fixed p151 --- spaces/S000036/properties/P000151.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/spaces/S000036/properties/P000151.md b/spaces/S000036/properties/P000151.md index e33b558bdb..b9b2504f20 100644 --- a/spaces/S000036/properties/P000151.md +++ b/spaces/S000036/properties/P000151.md @@ -1,7 +1,7 @@ --- space: S000036 property: P000151 -value: false +value: true --- -Player I can first pick the point $\omega_1 \in X$, then Player II has to pick a cocountauble neighborhood of $\omega_1 \in X$, giving Player I a winning strategy by {T341} and {T354}. +In the first round, after Player I picked some open cover, Player II can then pick a cocountable neighborhood of the point $\omega_1 \in X$. Player II then only has to cover countably many points, giving them a winning strategy by {T341} and {T354}. From fe2853a24b48619c1d3b9108bbf9c935bb09a002 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sat, 27 Dec 2025 22:25:24 +0100 Subject: [PATCH 7/7] update s36 p151 --- spaces/S000036/properties/P000151.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/spaces/S000036/properties/P000151.md b/spaces/S000036/properties/P000151.md index b9b2504f20..8cb181165a 100644 --- a/spaces/S000036/properties/P000151.md +++ b/spaces/S000036/properties/P000151.md @@ -4,4 +4,4 @@ property: P000151 value: true --- -In the first round, after Player I picked some open cover, Player II can then pick a cocountable neighborhood of the point $\omega_1 \in X$. Player II then only has to cover countably many points, giving them a winning strategy by {T341} and {T354}. +In the first round, after Player I picked some open cover, Player II can then pick a cocountable neighborhood $U$ of the point $\omega_1 \in X$. Choose an enumeration $X\setminus U = \{x_1,x_2,\dots\}$. Player II then has a winning strategy by picking a neighborhood of $x_n$ in the $n+1$-th step of the game.