definition of topological quasi-connectedness of 2 points
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of relation.
- The reader knows a definition of continuous, topological spaces map.
- The reader knows a definition of discrete topological space.
Target Context
- The reader will have a definition of topological quasi-connectedness of 2 points.
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:
\( T\): \(\in \{\text{ the topological spaces }\}\)
\( F\): \(= \{f: T \to T' \vert T' \in \{\text{ the discrete topological spaces }\}, f \in \{\text{ the continuous maps }\}\}\)
\(*R\): \(\in \{\text{ the relations on } T\}\)
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Conditions:
\(\forall t_1, t_2 \in T (t_1 R t_2 \iff \forall f \in F (f (t_1) = f (t_2)))\)
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2: Note
This is called "quasi-connected", because if \(t_1\) and \(t_2\) are connected, they are quasi-connected, but not necessarily vice versa, by the proposition that for any topological space, each connected component is contained in the corresponding quasi-connected component.
