pi-base · felixpernegger · Mar 13, 2026
diff --git 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
@@ -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
@@ -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
@@ -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
@@ -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
@@ -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 <https://www.jstor.org/stable/24901448>.
+
+----
+#### Meta-properties
+
+- This property is hereditary.
diff --git 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
@@ -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
@@ -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
@@ -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
@@ -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
@@ -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
@@ -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
@@ -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
@@ -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
@@ -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
@@ -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
@@ -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.
diff --git 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
@@ -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
@@ -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 <https://ncatlab.org/nlab/show/weakly+Hausdorff+topological+space>.
+
+----
+#### Meta-properties
+
+- This property is hereditary.
diff --git 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
@@ -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
@@ -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
@@ -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
@@ -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
@@ -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
@@ -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
@@ -11,3 +11,8 @@ refs:
 A space that is {P5} and {P183}.
 
 Originally defined in {{mr:206907}}, available at <https://www.jstor.org/stable/24901448>.
+
+----
+#### Meta-properties
+
+- This property is hereditary.
diff --git 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
@@ -32,5 +32,6 @@ See {{mr:206907}}, available at <https://www.jstor.org/stable/24901448>.
 ---
 ### 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
@@ -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
@@ -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
@@ -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
@@ -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.