diff --git a/properties/P000052.md b/properties/P000052.md index 0edad7f06..5bc00e50b 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 1a4e854d9..22b0ca48f 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 ed124d503..b3a372e77 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 b402a02dc..e60d288b6 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 f230feb52..5a3dbb25e 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 58a1f474a..58782fcb6 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 d6651778f..eb3ff4b61 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 d1ba32c57..4ab2e20bf 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 9cf6b8931..95158ac9e 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 465a6a03d..0c87a9e0f 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 2c4579557..6493d67d7 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 824520d71..939256fee 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 50bbab3e1..a0d386d66 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 1633d8469..25f035d14 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 0f65cd2fd..fd821f6ba 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 043685d5b..47021e463 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 62e68f81f..ef146e331 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 1f908b56a..d71271818 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 1e8f8250d..f031b26ac 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 1cb619c32..d965c9cb1 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 c5f101e08..579a7ea76 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 e46b72394..4dea65f1b 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 ec3550240..3d6d9c849 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 fb2d03922..553596999 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 8f73d61dd..af5a14586 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 a405c7e6a..0cb19c6f9 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 2bd44fbdf..d34d129cc 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 0b00ec98e..f25ee82dd 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 b5723dfb1..08be4d871 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 90311ae71..19e5cf03a 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 ee23d73c4..e5af30221 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 a0b0df360..b7c2d7537 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 6cdd0e311..848c0a93d 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 91bc4ebbf..d43021b72 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 222ee2349..72a0a9201 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.