Update S77 (Michael's product topology) (#1713)

yhx-12243 · web-flow · commit 993ca15a84f6 · 2026-03-31T14:08:43.000-04:00
diff --git a/properties/P000062.md b/properties/P000062.md
@@ -13,3 +13,4 @@ Every open cover of $X$ has a countable subcollection whose union is dense in $X
 
 - 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.
diff --git a/properties/P000166.md b/properties/P000166.md
@@ -17,4 +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)).
+- 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)).
diff --git a/properties/P000206.md b/properties/P000206.md
@@ -26,3 +26,4 @@ to the {{wikipedia:Normal_play_convention}} of combinatorial games.
 #### Meta-properties
 
 - $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
+- This property is preserved by arbitrary products and box products.
diff --git a/spaces/S000077/README.md b/spaces/S000077/README.md
@@ -1,9 +1,11 @@
 ---
 uid: S000077
-name: Michael's product topology
+name: Product of Michael line and irrational numbers
+aliases:
+  - Michael's product topology
 counterexamples_id: 85
 refs:
-  - zb: "0386.54001" 
+  - zb: "0386.54001"
     name: Counterexamples in Topology
 ---
 
diff --git a/spaces/S000077/properties/P000006.md b/spaces/S000077/properties/P000006.md
diff --git a/spaces/S000077/properties/P000028.md b/spaces/S000077/properties/P000028.md
@@ -0,0 +1,7 @@
+---
+space: S000077
+property: P000028
+value: true
+---
+
+{S63|P28} and {S28|P28}, hence so is their product.
diff --git a/spaces/S000077/properties/P000036.md b/spaces/S000077/properties/P000036.md
diff --git a/spaces/S000077/properties/P000050.md b/spaces/S000077/properties/P000050.md
@@ -0,0 +1,7 @@
+---
+space: S000077
+property: P000050
+value: true
+---
+
+{S63|P50} and {S28|P50}, hence so is their product.
diff --git a/spaces/S000077/properties/P000051.md b/spaces/S000077/properties/P000051.md
diff --git a/spaces/S000077/properties/P000062.md b/spaces/S000077/properties/P000062.md
@@ -0,0 +1,8 @@
+---
+space: S000077
+property: P000062
+value: false
+---
+
+If $X$ is {P62}, then so is each of its factors.
+However, {S63|P62}.
diff --git a/spaces/S000077/properties/P000063.md b/spaces/S000077/properties/P000063.md
@@ -0,0 +1,7 @@
+---
+space: S000077
+property: P000063
+value: false
+---
+
+$X$ contains {S63} as a closed subspace and {S63|P63}.
diff --git a/spaces/S000077/properties/P000065.md b/spaces/S000077/properties/P000065.md
diff --git a/spaces/S000077/properties/P000139.md b/spaces/S000077/properties/P000139.md
@@ -0,0 +1,8 @@
+---
+space: S000077
+property: P000139
+value: false
+---
+
+If $X$ {P139}, then so does each of its factors.
+However, {S28|P139}.
diff --git a/spaces/S000077/properties/P000166.md b/spaces/S000077/properties/P000166.md
@@ -0,0 +1,7 @@
+---
+space: S000077
+property: P000166
+value: true
+---
+
+The Euclidean topology of $\mathbb R \times (\mathbb R \setminus \mathbb Q)$ is a coarser topology which is {P26} and {P53}.
diff --git a/spaces/S000077/properties/P000167.md b/spaces/S000077/properties/P000167.md
diff --git a/spaces/S000077/properties/P000206.md b/spaces/S000077/properties/P000206.md
@@ -0,0 +1,7 @@
+---
+space: S000077
+property: P000206
+value: true
+---
+
+{S63|P206} and {S28|P206}, hence so is their product.

Original file line number	Diff line number	Diff line change
`@@ -13,3 +13,4 @@ Every open cover of $X$ has a countable subcollection whose union is dense in $X`
`13`	`13`
`14`	`14`	`- This property is hereditary with respect to clopen sets.`
`15`	`15`	`- This property is preserved in any coarser topology.`
	`16`	`+- If a nonempty product space satisfies the property, so does every factor.`
Original file line number	Diff line number	Diff line change
`@@ -26,3 +26,4 @@ to the {{wikipedia:Normal_play_convention}} of combinatorial games.`
`26`	`26`	`#### Meta-properties`
`27`	`27`
`28`	`28`	`- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.`
	`29`	`+- This property is preserved by arbitrary products and box products.`