diff --git a/theorems/T000895.md b/theorems/T000895.md new file mode 100644 index 000000000..d0019aae1 --- /dev/null +++ b/theorems/T000895.md @@ -0,0 +1,13 @@ +--- +uid: T000895 +if: + P000174: true +then: + P000210: true +--- + +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). +And {T748}. diff --git a/theorems/T000896.md b/theorems/T000896.md new file mode 100644 index 000000000..5775958a5 --- /dev/null +++ b/theorems/T000896.md @@ -0,0 +1,13 @@ +--- +uid: T000896 +if: + P000147: true +then: + P000210: true +--- + +To prove the result it is enough to show every countable subspace of $X$ is {P210}. +Note that each subspace of $X$ is also {P147}. +So wlog we may assume $X$ is {P57}, and hence {P28} +[(Explore)](https://topology.pi-base.org/spaces?q=Countable%2BP-space%2B%7EFirst+countable). +And {T748}. diff --git a/theorems/T000897.md b/theorems/T000897.md new file mode 100644 index 000000000..fa9f80147 --- /dev/null +++ b/theorems/T000897.md @@ -0,0 +1,13 @@ +--- +uid: T000897 +if: + P000187: true +then: + P000210: true +--- + +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}.