description/proof of that for topological group action, valid restriction of action w.r.t. subgroup topological subspace and topological subspace is topological group action
Topics
About: group
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological group left action.
- The reader knows a definition of topological group right action.
- The reader knows a definition of topological subspace.
- The reader admits the proposition that for any topological group, any subgroup as the topological subspace is a topological group.
- The reader admits the proposition that for any possibly uncountable number of indexed topological spaces or any finite number of topological spaces and their subspaces, the product of the subspaces is the subspace of the product of the base spaces.
Target Context
- The reader will have a description and a proof of the proposition that for any topological group left or right action, any valid restriction of the action with respect to any subgroup topological subspace and any topological subspace is a topological group left or right action.
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:
\( G'\): \(\in \{\text{ the topological groups }\}\)
\( T'\): \(\in \{\text{ the topological spaces }\}\)
\( f'_l\): \(: G' \times T' \to T'\), \(\in \{\text{ the topological group left actions }\}\)
\( f'_r\): \(: T' \times G' \to T'\), \(\in \{\text{ the topological group right actions }\}\)
\( G\): \(\in \{\text{ the subgroups of } G'\} \cap \{\text{ the topological subspaces of } G'\}\)
\( T\): \(\in \{\text{ the topological subspaces of } T'\}\)
//
Statements:
(
\(f_l: G \times T \to T, \text{ as the restriction of } f'_l, \text{ is valid }\)
\(\implies\)
\(f_l \in \{\text{ the topological group left actions }\}\)
)
\(\land\)
(
\(f_r: T \times G \to T, \text{ as the restriction of } f'_r, \text{ is valid }\)
\(\implies\)
\(f_r \in \{\text{ the topological group right actions }\}\)
)
//
"is valid" means that the map is really into the codomain.
2: Note
When \(T = T'\), \(f_l\) and \(f_r\) are certainly valid, so, \(f_l\) and \(f_r\) are some topological group actions.
3: Proof
Whole Strategy: Step 1: see that \(G\) is a topological group; Step 2: see that \(f_l\) and \(f_r\) are group actions; Step 3: see that \(f_l\) and \(f_r\) are continuous; Step 4: conclude the proposition.
Step 1:
\(G\) is a topological group, by the proposition that for any topological group, any subgroup as the topological subspace is a topological group.
Step 2:
Let us see that \(f_l\) is a group left action.
For each \(g_1, g_2 \in G\) and each \(t \in T\), \(f_l (g_2, f_l (g_1, t)) = f'_l (g_2, f'_l (g_1, t)) = f'_l (g_2 g_1, t)\), because \(f'_l\) is a group left action, \(= f_l (g_2 g_1, t)\).
For each \(t \in T\), \(f_l (1, t) = f'_l (1, t) = t\), because \(f'_l\) is a group left action.
So, \(f_l\) is a group left action.
Let us see that \(f_r\) is a group right action.
For each \(g_1, g_2 \in G\) and each \(t \in T\), \(f_r (f_r (t, g_1), g_2) = f'_r (f'_r (t, g_1), g_2) = f'_r (s, g_1 g_2)\), because \(f'_r\) is a group right action, \(= f_r (s, g_1 g_2)\).
For each \(t \in T\), \(f_r (t, 1) = f'_r (t, 1) = t\), because \(f'_r\) is a group right action.
So, \(f_r\) is a group right action.
Step 3:
\(f_l\) is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous: \(G \times T\) is a topological subspace of \(G' \times T'\), by the proposition that for any possibly uncountable number of indexed topological spaces or any finite number of topological spaces and their subspaces, the product of the subspaces is the subspace of the product of the base spaces.
\(f_r\) is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous: \(T \times G\) is a topological subspace of \(T' \times G'\), by the proposition that for any possibly uncountable number of indexed topological spaces or any finite number of topological spaces and their subspaces, the product of the subspaces is the subspace of the product of the base spaces.
Step 4:
So, \(f_l\) and \(f_r\) are a topological group left action and a topological group right action.
