Add "iff Kolmogorov ..." for most remaining properties

properties/P000010.md

@@ -14,6 +14,7 @@ Defined in 14E of {{zb:1052.54001}}.
 ----
 ### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to open sets.
 - This property is not hereditary with respect to closed sets
 (Example: {S11|P10}

properties/P000014.md

@@ -22,4 +22,5 @@ Defined on page 11 of {{zb:0386.54001}} (as $T_5$).
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary.

properties/P000015.md

@@ -31,4 +31,5 @@ Defined on page 16 of {{zb:0386.54001}} (as perfectly $T_4$).
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary (see Theorem 2.1.6 in {{zb:0684.54001}}).

properties/P000017.md

@@ -9,3 +9,8 @@ refs:
 A space which is the union of countably many {P16} subsets.
 
 Defined on page 19 of {{zb:0386.54001}}.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000019.md

@@ -19,4 +19,5 @@ Defined on page 19 of {{zb:0386.54001}}.  See for example {{wikipedia:Countably_
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to closed sets.

properties/P000020.md

@@ -15,5 +15,6 @@ Defined on page 19 of {{zb:0386.54001}}.
 ---
 ### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to closed sets.
 - This property is preserved by countable products (see Theorem 3.10.35 in {{zb:0684.54001}}).

properties/P000022.md

@@ -13,4 +13,5 @@ Defined on page 20 of {{zb:0386.54001}}.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is preserved in any coarser topology.

properties/P000024.md

@@ -16,3 +16,8 @@ Equivalently (see Condition 2 of {{wikipedia:Locally_compact_space}}), every poi
 a closed and compact neighborhood.
 
 Defined on page 20 of {{zb:0386.54001}} as "strongly locally compact". Contrast with {P130}.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000025.md

@@ -22,3 +22,8 @@ Equivalently, $X=\bigcup_{n<\omega}K_n$ for $K_n$ compact, and such that $K_n\su
 The equivalences above can be shown using the ideas in {{mathse:4568032}}.
 
 Defined on page 21 of {{zb:0386.54001}} as "$\sigma$-locally compact".
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000027.md

@@ -13,4 +13,5 @@ Defined on page 7 of {{zb:0386.54001}}.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary.

properties/P000029.md

@@ -17,3 +17,8 @@ A space in which every collection of pairwise-disjoint nonempty open sets is cou
 
 Defined on page 22 of {{zb:0386.54001}}.
 Also referred to as the "Suslin/Souslin property" as in e.g. 1.7.12 of {{zb:0684.54001}}.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000031.md

@@ -18,4 +18,5 @@ Defined on page 23 of {{zb:0386.54001}}.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to closed sets.

properties/P000032.md

@@ -13,4 +13,5 @@ Defined on page 23 of {{zb:0386.54001}}.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to closed sets.

properties/P000034.md

@@ -35,3 +35,8 @@ Also see [Henno Brandsma: On paracompactness, full normality and the like](http:
 
 (Note: {{zb:0386.54001}} defines this property on page 23 as "fully $T_4$".
 Misleadingly, it also calls "star refinement" what all other major sources call a barycentric refinement.)
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000039.md

@@ -24,6 +24,7 @@ See {{wikipedia:Hyperconnected_space}} for other equivalent characterizations.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to open sets.
 - This property is preserved by arbitrary products.
 - This property is preserved by continuous images.

properties/P000040.md

@@ -14,3 +14,8 @@ Equivalently ({{mathse:5006424}}), the only neighborhood of the space's
 diagonal $\Delta=\{\langle x,x\rangle:x\in X\}$ is $X^2$.
 
 Defined on page 29 of {{zb:0386.54001}}.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000041.md

@@ -32,4 +32,5 @@ Defined on page 30 of {{zb:0386.54001}}.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to open sets.

properties/P000042.md

@@ -44,4 +44,5 @@ Compare with {P43} and {P96}.
 ----
 ### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to open sets.

properties/P000049.md

@@ -26,6 +26,7 @@ which we do not assume here.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to open sets (see Problem 15G.2 in {{zb:1052.54001}}).
 - This property is hereditary with respect to dense sets (see {{mathse:3769214}}).
 - This property is hereditary with respect to locally dense sets (equivalent to previous two meta-properties; see also Proposition 1 of {{mathse:5025114}}).

properties/P000050.md

@@ -13,5 +13,6 @@ Defined on page 33 of {{zb:0386.54001}}.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary.
 - This property is preserved by arbitrary products.

properties/P000054.md

@@ -15,5 +15,6 @@ Defined on page 37 of {{zb:0386.54001}}.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary.
 - This property is preserved by arbitrary disjoint unions.

properties/P000056.md

@@ -17,4 +17,5 @@ Defined on page 7 of {{zb:0386.54001}}.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is preserved by arbitrary disjoint unions.

properties/P000060.md

@@ -13,3 +13,8 @@ refs:
 Every continuous function $f:X \to \mathbb R$ is constant.
 
 Defined on page 223 of {{zb:0386.54001}}.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000062.md

@@ -11,6 +11,7 @@ Every open cover of $X$ has a countable subcollection whose union is dense in $X
 ----
 ### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to clopen sets.
 - This property is preserved in any coarser topology.
 - If a nonempty product space satisfies the property, so does every factor.

properties/P000064.md

@@ -11,3 +11,8 @@ refs:
 A space such that the countable union of closed nowhere dense subsets has empty interior; equivalently, such that the countable intersection of open dense subsets is still dense.
 
 See {{wikipedia:Baire_space}} for several other equivalent characterizations.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000066.md

@@ -16,4 +16,5 @@ See section 5 of {{doi:10.1016/0166-8641(95)00067-4}}.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to closed sets.

properties/P000068.md

@@ -17,4 +17,5 @@ See section 6 of {{doi:10.1016/0166-8641(95)00067-4}}.
 
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to closed subspaces.

properties/P000069.md

@@ -20,4 +20,9 @@ Strategic Menger: The second player has a winning strategy in the Menger game. S
 
 Strategically $\Omega$-Menger: The second player has a winning strategy in the game $\mathsf{G}_{\mathrm{fin}}(\Omega_X,\Omega_X)$. See pages 2 and 3 of {{doi:10.1016/j.topol.2019.07.008}} for more details.
 
-The equivalence of Strategic Menger and Strategically $\Omega$-Menger is Theorem 35 of {{doi:10.1016/j.topol.2019.02.062}}.
+The equivalence of Strategic Menger and Strategically $\Omega$-Menger is Theorem 35 of {{doi:10.1016/j.topol.2019.02.062}}.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000070.md

@@ -17,4 +17,9 @@ Markov Menger: The second player has a Markov winning strategy in the Menger gam
 
 Markov $\Omega$-Menger: The second player has a Markov winning strategy in the game $\mathsf{G}_{\mathrm{fin}}(\Omega_X,\Omega_X)$ (relying on only the round number and most recent move of the opponent). See pages 2 and 3 of {{doi:10.1016/j.topol.2019.07.008}} for more details.
 
-The equivalence of Markov Menger with Markov $\Omega$-Menger is established in {{mathse:4730419}}.
+The equivalence of Markov Menger with Markov $\Omega$-Menger is established in {{mathse:4730419}}.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000079.md

@@ -17,6 +17,7 @@ Equivalently, for every set $A\subseteq X$ that is *not* closed in $X$, there is
 ----
 ### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to closed sets (see {{mathse:479808}}).
 - This property is hereditary with respect to open sets (see {{mathse:479808}}).
 - This property is preserved by quotient maps.

properties/P000080.md

@@ -17,5 +17,6 @@ Compare with {P172}.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary.
 

properties/P000090.md

@@ -27,6 +27,7 @@ See also {{zb:0944.54018}}.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary.
 - This property is preserved by arbitrary disjoint unions.
 - This property is preserved by finite products.

properties/P000098.md

@@ -22,3 +22,8 @@ its intersection with each $K_n$ is closed (resp. open) in $K_n$.
 Defined as "$k_\omega$-space" on page 13 of {{zb:0288.22006}}.
 The corresponding notion requiring that each $K_n$ be {P3} is {P92}.
 See also {P140}, {P141}, and {P142}.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000105.md

@@ -13,4 +13,5 @@ Defined on page 200 of {{zb:1059.54001}}.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is preserved by arbitrary disjoint unions.

properties/P000108.md

@@ -24,5 +24,6 @@ For the equivalence of the conditions above, see {{mathse:4737011}}.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary.
 - This property is preserved by arbitrary disjoint unions.

properties/P000117.md

@@ -15,4 +15,5 @@ See for example page 127 of {{zb:0684.54001}}.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary.

properties/P000128.md

@@ -8,4 +8,9 @@ refs:
 
 Every open $k$-cover of $X$ has a countable $k$-subcover.  (A family $\mathcal U$ of open subsets of $X$ is called a *$k$-cover* if each compact subset of $X$ is contained in an element of $\mathcal U$ and $X \notin \mathcal U$.)
 
-Defined on page 3279 of {{doi:10.1016/j.topol.2005.07.015}}.
+Defined on page 3279 of {{doi:10.1016/j.topol.2005.07.015}}.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000129.md

@@ -15,4 +15,5 @@ See 3.2d of {{zb:1052.54001}}.
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary.

properties/P000130.md

@@ -12,6 +12,7 @@ Given as condition (3) in {{wikipedia:Locally_compact_space}}.  See also the art
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to open sets.
 - This property is hereditary with respect to closed sets.
 - This property is hereditary with respect to locally closed sets (equivalent to previous two meta-properties).

properties/P000140.md

@@ -22,6 +22,7 @@ Equivalently, a space whose topology coincides with the final topology with resp
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to closed sets.
 - This property is not hereditary with respect to open sets
 (e.g., {S23}, open in {S165}).

properties/P000146.md

@@ -21,4 +21,5 @@ The equivalence between the two definitions is shown in Lemma 1.3 and Corollary
 ----
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is preserved by arbitrary disjoint unions.

properties/P000149.md

@@ -26,4 +26,5 @@ as the fifth item "$\varepsilon$" in a list from
 ---
 #### Meta-properties
 
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is hereditary with respect to closed sets.

properties/P000150.md

@@ -12,3 +12,8 @@ The space satisfies the selection principle $\mathsf S_1(\Omega,\Omega)$: for ev
 Equivalently, every finite power of $X$ is {P68}.
 
 See {{doi:10.1090/S0002-9939-1988-0964873-0}}; the equivalence with every finite power being {P68} is on page 918.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000152.md

@@ -21,3 +21,8 @@ Equivalent conditions:
 - Topologically countable: There is a set $\{ x_n : n \in \omega \} \subseteq X$ so that, for every $x \in X$, there is some $n \in \omega$ so that every neighborhood of $x_n$ contains $x$. See {{zb:1555.54009}} for more on this property.
 
 The equivalence is shown in Theorem 4.17 of {{zb:1555.54009}} and also at {{mathse:4737285}}.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000153.md

@@ -12,3 +12,8 @@ The space satisfies the selection principle $\mathsf S_{\mathrm{fin}}(\Omega,\Om
 Equivalently, every finite power of $X$ is {P66}.
 
 See {{doi:10.1016/S0166-8641(96)00075-2}}; the equivalence with every finite power being {P66} is Theorem 3.9.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000156.md

@@ -8,3 +8,8 @@ refs:
 
 The space satisfies the selection principle $\mathsf S_1(\mathcal K,\mathcal K)$: for every sequence $\langle \mathscr U_n : n \in \omega \rangle$ of $k$-covers of $X$, there exist choices $U_n \in \mathscr U_n$ so that $\{ U_n :n \in \omega \}$ is a $k$-cover of $X$.
 (A family $\mathcal U$ of open subsets of $X$ is called a *$k$-cover* if each compact subset of $X$ is contained in an element of $\mathcal U$ and $X \notin \mathcal U$.)
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000157.md

@@ -6,3 +6,8 @@ refs:
   name: Selection games and the Vietoris space
 ---
 The second player has a winning strategy in the game $\mathsf{G}_1(\mathcal K_X,\mathcal K_X)$. See Remark 2.4 and Definition 2.7 of {{doi:10.1016/j.topol.2021.107772}} for more details.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000159.md

@@ -8,3 +8,8 @@ refs:
 
 The space satisfies the selection principle $\mathsf S_{\mathrm{fin}}(\mathcal K,\mathcal K)$: for every sequence $\langle \mathscr U_n : n \in \omega \rangle$ of $k$-covers of $X$, there exist choices $\mathcal F_n$, a finite subset of $\mathscr U_n$, so that $\bigcup_{n\in\omega} \mathcal F_n$ is a $k$-cover of $X$.
 (A family $\mathcal U$ of open subsets of $X$ is called a *$k$-cover* if each compact subset of $X$ is contained in an element of $\mathcal U$ and $X \notin \mathcal U$.)
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000160.md

@@ -6,3 +6,8 @@ refs:
   name: Selection games and the Vietoris space
 ---
 The second player has a winning strategy in the game $\mathsf{G}_{\mathrm{fin}}(\mathcal K_X,\mathcal K_X)$. See Remark 2.4 and Definition 2.7 of {{doi:10.1016/j.topol.2021.107772}} for more details.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000161.md

@@ -5,4 +5,9 @@ refs:
 - doi: 10.1016/j.topol.2021.107772
   name: Selection games and the Vietoris space
 ---
-The second player has a Markov winning strategy in the game $\mathsf{G}_{\mathrm{fin}}(\mathcal K_X,\mathcal K_X)$ (relying on only the round number and most recent move of the opponent). See Remark 2.4 and Definition 2.7 of {{doi:10.1016/j.topol.2021.107772}} for more details.
+The second player has a Markov winning strategy in the game $\mathsf{G}_{\mathrm{fin}}(\mathcal K_X,\mathcal K_X)$ (relying on only the round number and most recent move of the opponent). See Remark 2.4 and Definition 2.7 of {{doi:10.1016/j.topol.2021.107772}} for more details.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.