2025-07-13

1201: Rough Section of Continuous Surjection

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definition of rough section of continuous surjection

Topics


About: topological space

The table of contents of this article


Starting Context



Target Context


  • The reader will have a definition of rough section of continuous surjection.

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:
\( T_1\): \(\in \{\text{ the topological spaces }\}\)
\( T_2\): \(\in \{\text{ the topological spaces }\}\)
\( \pi\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous surjections }\}\)
\(*s\): \(: T_2 \to T_1\)
//

Conditions:
\(\pi \circ s: T_2 \to T_2 = id\)
//

\(s\) is called "rough section of \(\pi\)".


3: Note


\(\pi\) needs to be surjective, because otherwise, there would be a \(t \in T_2\) that would not be mapped under \(\pi\) to, and then, \(\pi \circ s (t) = t\) would be impossible whatever \(s\) we chose, which means that \(\pi \circ s = id\) would be impossible.

Usually, we need only (non-rough) sections, but we sometimes need to talk about a rough section in order to 1st introduce a may-be-rough section and then prove that it is really a non-rough section.


References


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