definition of topological group
Topics
About: group
The table of contents of this article
Starting Context
- The reader knows a definition of Hausdorff topological space.
- The reader knows a definition of group.
- The reader knows a definition of continuous map.
Target Context
- The reader will have a definition of topological group.
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:
\( G\): \(\in \{\text{ the Hausdorff topological spaces }\} \cap \{\text{ the groups }\}\), with the group multiplication operation, \(f_1: G \times G \to G\), and the group inverse operation, \(f_2: G \to G\)
//
Conditions:
\(f_1 \in \{\text{ the continuous maps }\}\)
\(\land\)
\(f_2 \in \{\text{ the continuous maps }\}\)
//
2: Note
\(f_2\) is inevitably a homeomorphism, because \(f_2\) has the inverse, \(f_2\), which is continuous: \(f_2 \circ f_2 = id\), because for each \(g \in G\), \(f_2 \circ f_2 (g) = f_2 (g^{-1}) = {g^{-1}}^{-1} = g\).
