Connectedness is connected union-closed: Difference between revisions

Latest revision as of 18:09, 26 January 2012

This article gives the statement, and possibly proof, of a topological space property (i.e., connected space) satisfying a topological space metaproperty (i.e., connected union-closed property of topological spaces)
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Statement

Version using a pivoting set

Suppose $X$ is a topological space. Suppose $A$ is a subset of $X$ and $B_{i}, i \in I$ is a collection of subsets of $X$ . Suppose that:

$A$ is a connected space in the subspace topology.
For each $i \in I$ , $B_{i}$ is a connected space in the subspace topology.
$A \cap B_{i}$ is non-empty for each $i \in I$ .

Then, the space:

$A \cup ⋃_{i \in I} B_{i}$

is a connected space in the subspace topology from $X$ .

Version using finite hopping

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@@ Line 1: / Line 1: @@
+{{topospace metaproperty satisfaction|
+property = connected space|
+metaproperty = connected union-closed property of topological spaces}}
 ==Statement==
 ===Version using a pivoting set===
-Suppose <math>X<math> is a [[topological space]]. Suppose <math>A</math> is a subset of <math>X</math> and <matH>B_i, i \in I</math> is a collection of subsets of <math>X</matH>. Suppose that:
+Suppose <math>X</math> is a [[topological space]]. Suppose <math>A</math> is a subset of <math>X</math> and <matH>B_i, i \in I</math> is a collection of subsets of <math>X</math>. Suppose that:
 # <math>A</math> is a [[connected space]] in the [[subspace topology]].
 # For each <math>i \in I</math>, <math>B_i</math> is a [[connected space]] in the [[subspace topology]].
-# <math>A \cap B_i</math> is non-empty for each <math>i \in I</math>>
+# <math>A \cap B_i</math> is non-empty for each <math>i \in I</math>.
 Then, the space: