From 9af065d7179ba5b10718d039114cdeab039625f5 Mon Sep 17 00:00:00 2001 From: Batixx Date: Fri, 13 Mar 2026 04:04:35 +0100 Subject: [PATCH 1/2] add metaproperty --- properties/P000052.md | 5 +++++ properties/P000054.md | 1 + properties/P000057.md | 5 +++++ properties/P000058.md | 5 +++++ properties/P000059.md | 5 +++++ properties/P000074.md | 5 +++++ properties/P000078.md | 5 +++++ properties/P000082.md | 1 + properties/P000094.md | 5 +++++ properties/P000097.md | 5 +++++ properties/P000100.md | 5 +++++ properties/P000108.md | 1 + properties/P000112.md | 1 + properties/P000113.md | 5 +++++ properties/P000117.md | 5 +++++ properties/P000118.md | 1 + properties/P000129.md | 5 +++++ properties/P000131.md | 1 + properties/P000132.md | 1 + properties/P000137.md | 5 +++++ properties/P000143.md | 5 +++++ properties/P000144.md | 1 + properties/P000147.md | 1 + properties/P000163.md | 5 +++++ properties/P000164.md | 5 +++++ properties/P000171.md | 5 +++++ properties/P000177.md | 5 +++++ properties/P000178.md | 1 + properties/P000179.md | 5 +++++ properties/P000180.md | 5 +++++ properties/P000183.md | 1 + properties/P000184.md | 5 +++++ properties/P000185.md | 5 +++++ properties/P000220.md | 5 +++++ properties/P000222.md | 5 +++++ 35 files changed, 131 insertions(+) diff --git a/properties/P000052.md b/properties/P000052.md index 0edad7f06a..5bc00e50ba 100644 --- a/properties/P000052.md +++ b/properties/P000052.md @@ -9,3 +9,8 @@ refs: Every point is isolated, that is, $\{x\}$ is open for every point $x$. See 3.2c of {{zb:1052.54001}}. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000054.md b/properties/P000054.md index 1a4e854d98..22b0ca48f2 100644 --- a/properties/P000054.md +++ b/properties/P000054.md @@ -15,4 +15,5 @@ Defined on page 37 of {{zb:0386.54001}}. ---- #### Meta-properties +- This property is hereditary. - This property is preserved by arbitrary disjoint unions. diff --git a/properties/P000057.md b/properties/P000057.md index ed124d5033..b3a372e77e 100644 --- a/properties/P000057.md +++ b/properties/P000057.md @@ -12,3 +12,8 @@ refs: The cardinality of the space is less than or equal to the cardinality of $\mathbb N$. See sections 1.12-1.15 of {{zb:1052.54001}}. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000058.md b/properties/P000058.md index b402a02dc4..e60d288b64 100644 --- a/properties/P000058.md +++ b/properties/P000058.md @@ -12,3 +12,8 @@ refs: The cardinality of the space is less than the cardinality of $\mathbb R$. See sections 1.12-1.15 of {{zb:1052.54001}}. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000059.md b/properties/P000059.md index f230feb527..5a3dbb25e7 100644 --- a/properties/P000059.md +++ b/properties/P000059.md @@ -12,3 +12,8 @@ refs: The cardinality of the space is at most the cardinality of $\mathcal{P}(\mathbb R)$, the set of subsets of $\mathbb R$. See sections 1.12-1.15 of {{zb:1052.54001}}. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000074.md b/properties/P000074.md index 58a1f474ae..58782fcb6d 100644 --- a/properties/P000074.md +++ b/properties/P000074.md @@ -11,3 +11,8 @@ refs: A space that is {P5} and {P182}. The notion was originally defined in section 10 of {{mr:206907}}, which is available at . + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000078.md b/properties/P000078.md index d6651778fc..eb3ff4b61a 100644 --- a/properties/P000078.md +++ b/properties/P000078.md @@ -12,3 +12,8 @@ refs: The cardinality of the space is finite. See sections 1.12-1.15 of {{zb:1052.54001}}. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000082.md b/properties/P000082.md index d1ba32c573..4ab2e20bf7 100644 --- a/properties/P000082.md +++ b/properties/P000082.md @@ -13,4 +13,5 @@ Defined in exercise 7 of section 34 in {{zb:0951.54001}}. ---- #### Meta-properties +- This property is hereditary. - The Kolmogorov quotient $\text{Kol}(X)$ is {P82} iff $X$ is {P144}. diff --git a/properties/P000094.md b/properties/P000094.md index 9cf6b8931c..95158ac9eb 100644 --- a/properties/P000094.md +++ b/properties/P000094.md @@ -8,3 +8,8 @@ refs: name: Locally finite space on Wikipedia --- Every point has a finite neighborhood. Equivalently, the space has a basis of finite sets. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000097.md b/properties/P000097.md index 465a6a03dc..0c87a9e0ff 100644 --- a/properties/P000097.md +++ b/properties/P000097.md @@ -7,3 +7,8 @@ refs: --- Homeomorphic to a subspace of $\mathbb R$ ({S25}). + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000100.md b/properties/P000100.md index 2c45795575..6493d67d7a 100644 --- a/properties/P000100.md +++ b/properties/P000100.md @@ -9,3 +9,8 @@ refs: --- Every compact subset is closed. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000108.md b/properties/P000108.md index 824520d71d..939256feea 100644 --- a/properties/P000108.md +++ b/properties/P000108.md @@ -24,4 +24,5 @@ For the equivalence of the conditions above, see {{mathse:4737011}}. ---- #### Meta-properties +- This property is hereditary. - This property is preserved by arbitrary disjoint unions. diff --git a/properties/P000112.md b/properties/P000112.md index 50bbab3e10..a0d386d663 100644 --- a/properties/P000112.md +++ b/properties/P000112.md @@ -16,4 +16,5 @@ Equivalently, there is a continuous injective map from $X$ into some metrizable ---- #### Meta-properties +- This property is hereditary. - This property is preserved by arbitrary disjoint unions. diff --git a/properties/P000113.md b/properties/P000113.md index 1633d84694..25f035d149 100644 --- a/properties/P000113.md +++ b/properties/P000113.md @@ -9,3 +9,8 @@ refs: A space that is {P5} and {P110}. See Definition 23.6 in {{zb:1052.54001}}. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000117.md b/properties/P000117.md index 0f65cd2fdb..fd821f6ba4 100644 --- a/properties/P000117.md +++ b/properties/P000117.md @@ -11,3 +11,8 @@ A space with a $\sigma$-locally finite network. A family $\mathcal N$ of subsets of $X$ is called a *network* if every open set is the union of a subfamily of $\mathcal N$; that is, for every open set $U$ and point $x\in U$ there is some $A\in\mathcal N$ with $x\in A\subseteq U$. The family $\mathcal N$ is *$\sigma$-locally finite* if it is a countable union of locally finite families. See for example page 127 of {{zb:0684.54001}}. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000118.md b/properties/P000118.md index 043685d5b9..47021e4632 100644 --- a/properties/P000118.md +++ b/properties/P000118.md @@ -15,4 +15,5 @@ See for example Definition 2.1 in {{doi:10.1016/j.topol.2015.05.085}}. ---- #### Meta-properties +- This property is hereditary. - $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does. diff --git a/properties/P000129.md b/properties/P000129.md index 62e68f81fc..ef146e3315 100644 --- a/properties/P000129.md +++ b/properties/P000129.md @@ -11,3 +11,8 @@ refs: The only open sets are $\emptyset$ and $X$. See 3.2d of {{zb:1052.54001}}. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000131.md b/properties/P000131.md index 1f908b56a0..d712718182 100644 --- a/properties/P000131.md +++ b/properties/P000131.md @@ -19,4 +19,5 @@ Defined in 16E of {{zb:1052.54001}}. ---- #### Meta-properties +- This property is hereditary. - $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does. \ No newline at end of file diff --git a/properties/P000132.md b/properties/P000132.md index 1e8f8250dc..f031b26acc 100644 --- a/properties/P000132.md +++ b/properties/P000132.md @@ -22,4 +22,5 @@ Note: A $G_\delta$ space is sometimes called a "perfect space" (Exercise 1.5.H(a ---- #### Meta-properties +- This property is hereditary. - $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does. diff --git a/properties/P000137.md b/properties/P000137.md index 1cb619c323..d965c9cb1a 100644 --- a/properties/P000137.md +++ b/properties/P000137.md @@ -9,3 +9,8 @@ refs: --- A space with no element, necessarily equal to {S163}. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000143.md b/properties/P000143.md index c5f101e089..579a7ea767 100644 --- a/properties/P000143.md +++ b/properties/P000143.md @@ -13,3 +13,8 @@ refs: A space $X$ such that for every compact Hausdorff space $K$ and every continuous map $f:K\to X$, the image $f(K)$ is closed in $X$. Originally defined on p. 275 of {{doi:10.1090/S0002-9947-1969-0251719-4}}. See also . + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000144.md b/properties/P000144.md index e46b72394e..4dea65f1b8 100644 --- a/properties/P000144.md +++ b/properties/P000144.md @@ -11,4 +11,5 @@ Every point has a neighborhood that is {P121}. ---- #### Meta-properties +- This property is hereditary. - $X$ is {P144} iff its Kolmogorov quotient $\text{Kol}(X)$ is {P82}. diff --git a/properties/P000147.md b/properties/P000147.md index ec35502406..3d6d9c8490 100644 --- a/properties/P000147.md +++ b/properties/P000147.md @@ -17,4 +17,5 @@ Defined in {{doi:10.1016/0016-660X(72)90026-8}} with the additional assumption o ---- #### Meta-properties +- This property is hereditary. - $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does. diff --git a/properties/P000163.md b/properties/P000163.md index fb2d03922d..5535969991 100644 --- a/properties/P000163.md +++ b/properties/P000163.md @@ -4,3 +4,8 @@ name: Cardinality $\leq\mathfrak c$ --- The cardinality of the space is less than or equal to the cardinality of $\mathbb R$. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000164.md b/properties/P000164.md index 8f73d61dd3..af5a145864 100644 --- a/properties/P000164.md +++ b/properties/P000164.md @@ -26,3 +26,8 @@ Note: Some authors, for example {{doi:10.1007/978-1-4615-7819-2}}, refer to meas (The existence of a measurable cardinal cannot be proven in ZFC. So spaces whose construction does not depend on set-theoretic axioms beyond ZFC should never have this property marked as false.) + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000171.md b/properties/P000171.md index a405c7e6a2..0cb19c6f9d 100644 --- a/properties/P000171.md +++ b/properties/P000171.md @@ -26,3 +26,8 @@ reference to this weakening of {P3}. Note: this property is not to be confused with other variants of k-Hausdorff (e.g. {P170}), where k-closed is defined in terms of different notions of compactly generated space or k-space. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000177.md b/properties/P000177.md index 2bd44fbdf7..d34d129cc6 100644 --- a/properties/P000177.md +++ b/properties/P000177.md @@ -11,3 +11,8 @@ refs: A space that is {P5} and {P117}. The name "$\sigma$-space" was introduced by A. Okuyama (see {{doi:10.3792/pja/1195521153}}) to denote only the {P117} property. But most later papers, for example {{doi:10.1016/j.topol.2015.05.085}}, also include {P5} in this definition. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000178.md b/properties/P000178.md index 0b00ec98e4..f25ee82dde 100644 --- a/properties/P000178.md +++ b/properties/P000178.md @@ -15,4 +15,5 @@ The notion was introduced in {{doi:10.1090/S0002-9939-1971-0276919-3}}. ---- #### Meta-properties +- This property is hereditary. - The Kolmogorov quotient $\text{Kol}(X)$ is {P178} iff $X$ is {P11} and {P118}. diff --git a/properties/P000179.md b/properties/P000179.md index b5723dfb17..08be4d8713 100644 --- a/properties/P000179.md +++ b/properties/P000179.md @@ -11,3 +11,8 @@ refs: A space that is {P5} and {P183}. Originally defined in {{mr:206907}}, available at . + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000180.md b/properties/P000180.md index 90311ae710..19e5cf03ad 100644 --- a/properties/P000180.md +++ b/properties/P000180.md @@ -9,3 +9,8 @@ refs: A space for which every subspace is {P26}. Defined on page 182 of {{doi:10.1016/B978-0-444-50355-8.X5000-4}}. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000183.md b/properties/P000183.md index ee23d73c44..e5af302216 100644 --- a/properties/P000183.md +++ b/properties/P000183.md @@ -32,5 +32,6 @@ See {{mr:206907}}, available at . --- ### Meta-properties +- This property is hereditary. - This property is preserved by countable products (see Corollary in {{mo:506308}}). diff --git a/properties/P000184.md b/properties/P000184.md index a0b0df360a..b7c2d75378 100644 --- a/properties/P000184.md +++ b/properties/P000184.md @@ -11,3 +11,8 @@ refs: Homeomorphic to a subspace of $\mathbb R^n$ for some finite $n$. Per Theorems 1.1.2 and 1.11.4 of {{zb:0401.54029}}, this is equivalent to being {P27}, {P53}, and finite-dimensional, using any of the small inductive, large inductive, or covering dimensions by Theorem 1.7.7. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000185.md b/properties/P000185.md index 6cdd0e3113..848c0a93d5 100644 --- a/properties/P000185.md +++ b/properties/P000185.md @@ -21,3 +21,8 @@ For proof of the equivalences and further characterizations, see section 13 of { Defined as example #5 ("Partition Topology") in {{zb:0386.54001}}. + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000220.md b/properties/P000220.md index 91bc4ebbf7..d43021b724 100644 --- a/properties/P000220.md +++ b/properties/P000220.md @@ -11,3 +11,8 @@ There exists an ultrametric inducing the topology of $X$. A metric $d$ is called an *ultrametric* if it satisfies a property stronger than the triangle inequality, namely: $\quad d(x, z) \leq \max(d(x, y), d(y, z)).$ + +---- +#### Meta-properties + +- This property is hereditary. diff --git a/properties/P000222.md b/properties/P000222.md index 222ee2349a..72a0a92019 100644 --- a/properties/P000222.md +++ b/properties/P000222.md @@ -12,3 +12,8 @@ Equivalently, any of the following: - A set $A\subseteq X$ is closed iff $A$ is finite or $A=X$. - $X$ is *minimal $T_1$*, i.e., {P2} with no strictly coarser {P2} topology. + +---- +#### Meta-properties + +- This property is hereditary. From 54101eee0acc0f0a294f97f34aee75ca4afb6cd3 Mon Sep 17 00:00:00 2001 From: Batixx Date: Thu, 19 Mar 2026 01:56:44 +0100 Subject: [PATCH 2/2] p166 is separable --- properties/P000166.md | 1 + 1 file changed, 1 insertion(+) diff --git a/properties/P000166.md b/properties/P000166.md index 25ee2f73a9..37f233b094 100644 --- a/properties/P000166.md +++ b/properties/P000166.md @@ -17,3 +17,4 @@ the function space $C_p(X)$ being {P26}. #### Meta-properties - This property is preserved by countable disjoint unions. +- This property is hereditary. (use the fact that [Metrizable + Separable => Hereditarily separable](https://topology.pi-base.org/spaces?q=metrizable+%2B+Separable+%2B+%7EHereditarily+separable))