P247: Has connected quasicomponents

properties/P000047.md

@@ -20,3 +20,4 @@ Some authors (for example {{zb:0684.54001}}) use "totally disconnected" to mean
 #### Meta-properties
 
 - This property is hereditary.
+- An arbitrary product of nonempty spaces satisfies this property iff each of its factors does.

properties/P000048.md

@@ -8,11 +8,15 @@ refs:
     name: Counterexamples in Topology
   - zb: "0684.54001"
     name: General Topology (Engelking, 1989)
+  - mathse: 3642317
+    name: The equivalence relation induced by the partition into quasi-components is multiplicative
 ---
 
 Given any two distinct points $x,y \in X$, there exists a clopen subset $A\subseteq X$ such that
 $x\in A$ and $y\not\in A$.
 
+Equivalently, every quasicomponent of $X$ is a singleton.
+
 Defined on page 32 of {{zb:0386.54001}}.
 
 Some authors (for example {{zb:0684.54001}}) use "totally disconnected" to mean {P48} and "hereditarily disconnected" to mean {P47}.
@@ -21,3 +25,6 @@ Some authors (for example {{zb:0684.54001}}) use "totally disconnected" to mean
 #### Meta-properties
 
 - This property is hereditary.
+- An arbitrary product of nonempty spaces satisfies this property iff each of its factors does.$^{[1]}$
+
+[1] use: the quasicomponents are products of quasicomponents in each factor; see {{mathse:3642317}}.

properties/P000050.md

@@ -17,5 +17,6 @@ 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/P000247.md

@@ -0,0 +1,27 @@
+---
+uid: P000247
+name: Has connected quasicomponents
+refs:
+  - zb: "0684.54001"
+    name: General Topology (Engelking, 1989)
+  - mathse: 3642317
+    name: The equivalence relation induced by the partition into quasi-components is multiplicative
+---
+
+Every quasicomponent of $X$ is connected.
+That is, quasicomponents and connected components coincide.
+
+Equivalently, any two distinct connected components $A$ and $B$ of $X$ can be separated by clopen sets
+(i.e., $A\subseteq U$ and $B\subseteq X\setminus U$ for some clopen set $U$).
+
+The *quasicomponent* of a point $x\in X$ is the intersection of all clopen sets containing $x$.
+It is the set of point $y\in X$ that cannot be separated from $x$ by a clopen set.
+See definition on p. 356 of {{zb:0684.54001}}.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
+- An arbitrary product of nonempty spaces satisfies this property iff each of its factors does.$^{[1]}$
+
+[1] use: the quasicomponents are products of quasicomponents in each factor; see {{mathse:3642317}}.

theorems/T000932.md

@@ -0,0 +1,9 @@
+---
+uid: T000932
+if:
+  P000234: true
+then:
+  P000247: true
+---
+
+Every connected component is clopen, hence equal to its quasicomponent.

theorems/T000933.md

@@ -0,0 +1,14 @@
+---
+uid: T000933
+if:
+  and:
+  - P000016: true
+  - P000003: true
+then:
+  P000247: true
+refs:
+  - zb: "0684.54001"
+    name: General Topology (Engelking, 1989)
+---
+
+See Theorem 6.23 of {{zb:0684.54001}}.

theorems/T000934.md

@@ -0,0 +1,9 @@
+---
+uid: T000934
+if:
+  P000048: true
+then:
+  P000247: true
+---
+
+Immediate from the definitions.

theorems/T000935.md

@@ -0,0 +1,11 @@
+---
+uid: T000935
+if:
+  P000050: true
+then:
+  P000247: true
+---
+
+Follows by passing to the Kolmogorov quotient,
+since any {P50} {P1} space {P247}
+[(Explore)](https://topology.pi-base.org/spaces?q=Zero+dimensional%2BT0%2B%7EHas+connected+quasicomponents).

theorems/T000936.md

@@ -0,0 +1,11 @@
+---
+uid: T000936
+if:
+  and:
+  - P000047: true
+  - P000247: true
+then:
+  P000048: true
+---
+
+Immediate from the definitions.