diff --git a/properties/P000210.md b/properties/P000210.md index 3f21fdc5fd..7ae5540a4a 100644 --- a/properties/P000210.md +++ b/properties/P000210.md @@ -30,3 +30,8 @@ as done in {{zb:0774.54019}}. The $\alpha_i$ properties for $i = 1, 2, 3, 4$ (sometimes called "sheaf amalgamation properties") are due to Arkhangel'skii ({{zb:0275.54004}}). + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000211.md b/properties/P000211.md index 5ba213be72..45bdf65186 100644 --- a/properties/P000211.md +++ b/properties/P000211.md @@ -29,3 +29,8 @@ One can also phrase the definition in terms of (injective) sequences of elements as done in {{zb:0774.54019}}. This property was introduced by Nyikos in {{zb:0774.54019}}. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000212.md b/properties/P000212.md index e904042ba3..0489e2f72f 100644 --- a/properties/P000212.md +++ b/properties/P000212.md @@ -29,3 +29,8 @@ as done in {{zb:0774.54019}}. The $\alpha_i$ properties for $i = 1, 2, 3, 4$ (sometimes called "sheaf amalgamation properties") are due to Arkhangel'skii ({{zb:0275.54004}}). + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000213.md b/properties/P000213.md index 89e031feac..8e5da280e1 100644 --- a/properties/P000213.md +++ b/properties/P000213.md @@ -29,3 +29,8 @@ as done in {{zb:0774.54019}}. The $\alpha_i$ properties for $i = 1, 2, 3, 4$ (sometimes called "sheaf amalgamation properties") are due to Arkhangel'skii ({{zb:0275.54004}}). + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000214.md b/properties/P000214.md index 394872eef2..47a4b1e41b 100644 --- a/properties/P000214.md +++ b/properties/P000214.md @@ -29,3 +29,8 @@ as done in {{zb:0774.54019}}. The $\alpha_i$ properties for $i = 1, 2, 3, 4$ (sometimes called "sheaf amalgamation properties") are due to Arkhangel'skii ({{zb:0275.54004}}). + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/spaces/S000031/properties/P000036.md b/spaces/S000031/properties/P000036.md deleted file mode 100644 index d07717707c..0000000000 --- a/spaces/S000031/properties/P000036.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000031 -property: P000036 -value: true -refs: -- zb: "1052.54001" - name: General Topology (Willard) ---- - -The space {S29} has the {P36} property. And the property is preserved by products, as shown in Theorem 26.10 of {{zb:1052.54001}}. diff --git a/spaces/S000031/properties/P000084.md b/spaces/S000031/properties/P000084.md index 01947370d4..de90eb11b0 100644 --- a/spaces/S000031/properties/P000084.md +++ b/spaces/S000031/properties/P000084.md @@ -4,4 +4,4 @@ property: P000084 value: false --- -It has {S29} as a subspace and {S29|P84}. +$X$ has {S29} as a subspace and {S29|P84}. diff --git a/spaces/S000031/properties/P000100.md b/spaces/S000031/properties/P000100.md deleted file mode 100644 index 5ab9bac341..0000000000 --- a/spaces/S000031/properties/P000100.md +++ /dev/null @@ -1,12 +0,0 @@ ---- -space: S000031 -property: P000100 -value: false -refs: -- mathse: 632320 - name: Weak Hausdorff space not KC ---- - -The diagonal $\Delta=\{(x,x):x\in\mathbb Q^*\}\subseteq X$ is homeomorphic to $\mathbb Q^*$ ({S29}), which is {P16}. But it is not closed in $X=\mathbb Q^*\times\mathbb Q^*$ since $\mathbb Q^*$ is not {P3}. This shows that $X$ is not {P100}. - -See {{mathse:632320}}. diff --git a/spaces/S000031/properties/P000138.md b/spaces/S000031/properties/P000138.md index 85820d1516..d32b63cfc5 100644 --- a/spaces/S000031/properties/P000138.md +++ b/spaces/S000031/properties/P000138.md @@ -4,4 +4,5 @@ property: P000138 value: false --- -Since {S29|P138}, there are uncountably many continuous maps $X\to X$. +For any continuous $f:\mathbb{Q}^*\to \mathbb{Q}^*$, $f\times f: X \to X$ is also continuous. +As {S29|P138}, this gives us uncountably many continuous self-maps of $X$. diff --git a/spaces/S000031/properties/P000214.md b/spaces/S000031/properties/P000214.md new file mode 100644 index 0000000000..12ae397f05 --- /dev/null +++ b/spaces/S000031/properties/P000214.md @@ -0,0 +1,7 @@ +--- +space: S000031 +property: P000214 +value: false +--- + +$X$ has {S29} as a subspace and {S29|P214}. diff --git a/spaces/S000031/properties/P000216.md b/spaces/S000031/properties/P000216.md new file mode 100644 index 0000000000..dab3e34cea --- /dev/null +++ b/spaces/S000031/properties/P000216.md @@ -0,0 +1,7 @@ +--- +space: S000031 +property: P000216 +value: false +--- + +$X$ has {S29} as a subspace and {S29|P216}.