From b127df5bf3968940ad882173b42975d55b34f524 Mon Sep 17 00:00:00 2001 From: Batixx Date: Tue, 19 May 2026 10:40:08 +0200 Subject: [PATCH 1/5] alpha beta gamma --- theorems/T000895.md | 9 +++++++++ theorems/T000896.md | 9 +++++++++ theorems/T000897.md | 9 +++++++++ 3 files changed, 27 insertions(+) create mode 100644 theorems/T000895.md create mode 100644 theorems/T000896.md create mode 100644 theorems/T000897.md diff --git a/theorems/T000895.md b/theorems/T000895.md new file mode 100644 index 000000000..ce1d4f36f --- /dev/null +++ b/theorems/T000895.md @@ -0,0 +1,9 @@ +--- +uid: T000895 +if: + P000174: true +then: + P000210: true +--- + +Utilising the metaproperties of {P210} and {P174}, we may assume that $X$ is {P57}. But then the assertion follows from already known theorems ([Explore](https://topology.pi-base.org/spaces?q=Countable+%2B+Well-based+%2B+%7E%24%5Calpha_1%24)). diff --git a/theorems/T000896.md b/theorems/T000896.md new file mode 100644 index 000000000..53ded205b --- /dev/null +++ b/theorems/T000896.md @@ -0,0 +1,9 @@ +--- +uid: T000896 +if: + P000147: true +then: + P000210: true +--- + +Utilising the metaproperties of {P210} and {P147}, we may assume that $X$ is {P57}. But then the assertion follows from already known theorems ([Explore](https://topology.pi-base.org/spaces?q=Countable+%2B+P-space+%2B+%7E%24%5Calpha_1%24)). diff --git a/theorems/T000897.md b/theorems/T000897.md new file mode 100644 index 000000000..1e88963a6 --- /dev/null +++ b/theorems/T000897.md @@ -0,0 +1,9 @@ +--- +uid: T000897 +if: + P000187: true +then: + P000210: true +--- + +Utilising the metaproperties of {P210} and {P187}, we may assume that $X$ is {P57}. But then the assertion follows from already known theorems ([Explore](https://topology.pi-base.org/spaces?q=Countable+%2B+W-space+%2B+%7E%24%5Calpha_1%24)). From 64446abc7c755a47e5cc23009b0a3f56c4b40fcf Mon Sep 17 00:00:00 2001 From: Batixx Date: Tue, 19 May 2026 10:49:35 +0200 Subject: [PATCH 2/5] write down explicitly --- theorems/T000895.md | 2 +- theorems/T000896.md | 2 +- theorems/T000897.md | 2 +- 3 files changed, 3 insertions(+), 3 deletions(-) diff --git a/theorems/T000895.md b/theorems/T000895.md index ce1d4f36f..f28b48dbb 100644 --- a/theorems/T000895.md +++ b/theorems/T000895.md @@ -6,4 +6,4 @@ then: P000210: true --- -Utilising the metaproperties of {P210} and {P174}, we may assume that $X$ is {P57}. But then the assertion follows from already known theorems ([Explore](https://topology.pi-base.org/spaces?q=Countable+%2B+Well-based+%2B+%7E%24%5Calpha_1%24)). +Utilising the metaproperties of {P210} and {P174}, we may assume that $X$ is {P57}. But then the assertion follows from {T238}, {T748}, {T186} and {T217}. diff --git a/theorems/T000896.md b/theorems/T000896.md index 53ded205b..c776dbefa 100644 --- a/theorems/T000896.md +++ b/theorems/T000896.md @@ -6,4 +6,4 @@ then: P000210: true --- -Utilising the metaproperties of {P210} and {P147}, we may assume that $X$ is {P57}. But then the assertion follows from already known theorems ([Explore](https://topology.pi-base.org/spaces?q=Countable+%2B+P-space+%2B+%7E%24%5Calpha_1%24)). +Utilising the metaproperties of {P210} and {P147}, we may assume that $X$ is {P57}. But then the assertion follows from {T238}, {T748}, {T894} and {T285}. diff --git a/theorems/T000897.md b/theorems/T000897.md index 1e88963a6..66a587aac 100644 --- a/theorems/T000897.md +++ b/theorems/T000897.md @@ -6,4 +6,4 @@ then: P000210: true --- -Utilising the metaproperties of {P210} and {P187}, we may assume that $X$ is {P57}. But then the assertion follows from already known theorems ([Explore](https://topology.pi-base.org/spaces?q=Countable+%2B+W-space+%2B+%7E%24%5Calpha_1%24)). +Utilising the metaproperties of {P210} and {P187}, we may assume that $X$ is {P57}. But then the assertion follows from {T238}, {T580} and {T748}. From 6c8c1da9a8fa5182b14b65355ff6fb3435f5b6b6 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Thu, 21 May 2026 06:46:19 +0200 Subject: [PATCH 3/5] Update theorems/T000895.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- theorems/T000895.md | 6 +++++- 1 file changed, 5 insertions(+), 1 deletion(-) diff --git a/theorems/T000895.md b/theorems/T000895.md index f28b48dbb..f065d2d4a 100644 --- a/theorems/T000895.md +++ b/theorems/T000895.md @@ -6,4 +6,8 @@ then: P000210: true --- -Utilising the metaproperties of {P210} and {P174}, we may assume that $X$ is {P57}. But then the assertion follows from {T238}, {T748}, {T186} and {T217}. +To prove the result it is enough to show each of the countable subspaces of $X$ is {P210}. +Note that each subspace of $X$ is also {P174}. +So wlog we may assume $X$ is {P57}, and hence {P28} +[(Explore)](https://topology.pi-base.org/spaces?q=Countable%2BWell-based%2B%7EFirst+countable). +And {T748}. From 3f943761e9e914c525f21fbeaec53e82515168cc Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Fri, 22 May 2026 08:24:44 +0200 Subject: [PATCH 4/5] Update theorems/T000897.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- theorems/T000897.md | 6 +++++- 1 file changed, 5 insertions(+), 1 deletion(-) diff --git a/theorems/T000897.md b/theorems/T000897.md index 66a587aac..fa9f80147 100644 --- a/theorems/T000897.md +++ b/theorems/T000897.md @@ -6,4 +6,8 @@ then: P000210: true --- -Utilising the metaproperties of {P210} and {P187}, we may assume that $X$ is {P57}. But then the assertion follows from {T238}, {T580} and {T748}. +To prove the result it is enough to show every countable subspace of $X$ is {P210}. +Note that each subspace of $X$ is also {P187}. +So wlog we may assume $X$ is {P57}, and hence {P28} +[(Explore)](https://topology.pi-base.org/spaces?q=Countable%2BW-space%2B%7EFirst+countable). +And {T748}. From edf4b6ac0c35d154b58dc8c98d81d804cbed9734 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Fri, 22 May 2026 08:24:53 +0200 Subject: [PATCH 5/5] Update theorems/T000895.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- theorems/T000895.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000895.md b/theorems/T000895.md index f065d2d4a..d0019aae1 100644 --- a/theorems/T000895.md +++ b/theorems/T000895.md @@ -6,7 +6,7 @@ then: P000210: true --- -To prove the result it is enough to show each of the countable subspaces of $X$ is {P210}. +To prove the result it is enough to show every countable subspace of $X$ is {P210}. Note that each subspace of $X$ is also {P174}. So wlog we may assume $X$ is {P57}, and hence {P28} [(Explore)](https://topology.pi-base.org/spaces?q=Countable%2BWell-based%2B%7EFirst+countable).