description/proof of that domain restriction of open map is not necessarily 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 a domain restriction of an open map is not necessarily 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:
\(T'_1\): \(\in \{\text{ the topological spaces }\}\)
\(T_2\): \(\in \{\text{ the topological spaces }\}\)
\(f'\): \(: T'_1 \to T_2\), \(\in \{\text{ the open maps }\}\)
\(T_1\): \(\in \{\text{ the topological subspaces of } T'_1\}\)
\(f\): \(: T_1 \to T_2, t_1 \mapsto f' (t_1)\)
//
Statements:
not necessarily \(f \in \{\text{ the open maps }\}\)
//
2: Proof
Whole Strategy: Step 1: see a counterexample.
Step 1:
Let \(T'_1 = \mathbb{R}\), the Euclidean topological space, \(T_1 = [0, 1) \subseteq T'_1\), \(T_2 = \mathbb{R}\), the Euclidean topological space, and \(f' = id\).
\(f'\) is an open map, because it is the identity map.
\([0, 1) \subseteq T_1\) is open on \(T_1\), but \(f ([0, 1)) = [0, 1) \subseteq T_2\) is not open on \(T_2\).
3: Note
Compare with the proposition that the domain restriction of any open map on any open subspace is open.
