description/proof of that topological quasi-connectedness of 2 points is equivalence relation on topological space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological quasi-connectedness of 2 points.
- The reader knows a definition of equivalence relation on set.
Target Context
- The reader will have a description and a proof of the proposition that the topological quasi-connectedness of 2 points is an equivalence relation on any topological space.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(T\): \(\in \{\text{ the topological spaces }\}\)
\(R\): \(= \text{ the quasi-connectedness of } 2 \text{ points on } T\)
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Statements:
\(R \in \{\text{ the equivalence relations }\}\)
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2: Proof
Whole Strategy: Step 1: see that \(R\) satisfies the conditions to be an equivalence relation.
Step 1:
1) \(\forall t \in T (t \sim t)\): reflexivity: for each \(f \in F\), \(f (t) = f (t)\).
2) \(\forall t_1, t_2 \in T (t_1 \sim t_2 \implies t_2 \sim t_1)\): symmetry: for each \(f \in F\), \(f (t_1) = f (t_2)\), so, for each \(f \in F\), \(f (t_2) = f (t_1)\).
3) \(\forall t_1, t_2, t_3 \in T ((t_1 \sim t_2 \land t_2 \sim t_3)\implies t_1 \sim t_3)\): transitivity: for each \(f \in F\), \(f (t_1) = f (t_2)\) and \(f (t_2) = f (t_3)\), so, for each \(f \in F\), \(f (t_1) = f (t_3)\).
So, \(R\) is an equivalence relation.
