description/proof of that codomain restriction of open map is open
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of open map.
- The reader knows a definition of topological subspace.
Target Context
- The reader will have a description and a proof of the proposition that any codomain restriction of any open map is open.
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:
\(T_1\): \(\in \{\text{ the topological spaces }\}\)
\(T'_2\): \(\in \{\text{ the topological spaces }\}\)
\(f'\): \(: T_1 \to T'_2\), \(\in \{\text{ the open maps }\}\)
\(T_2\): \(\in \{\text{ the topological subspaces of } T'_2\}\) such that \(f' (T_1) \subseteq T_2\)
\(f\): \(: T_1 \to T_2, t_1 \mapsto f' (t_1)\)
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Statements:
\(f \in \{\text{ the open maps }\}\)
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2: Proof
Whole Strategy: Step 1: for each open subset, \(U_1 \subseteq T_1\), see that \(f (U_1) \subseteq T_2\) is open.
Step 1:
Let \(U_1 \subseteq T_1\) be any open subset.
\(f' (U_1) \subseteq T'_2\) is an open subset of \(T'_2\).
\(f (U_1) = f' (U_1) = f' (U_1) \cap T_2\), which is open on \(T_2\).
3: Note
Compare with the proposition that a codomain extension of an open map is not necessarily open.
