1189: Topological Group

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definition of topological group

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Topics

About: group

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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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.

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Target Context

• The reader will have a definition of topological group.

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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.

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Main Body

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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, $f_2:G\to G$
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Conditions:
$f_1\in \{\text{ the continuous maps }\}$
$\wedge$
$f_2\in \{\text{ the continuous maps }\}$
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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 $g\in G$, $f_2\circ f_2(g)=f_2(g^{-1})={g^{-1}}^{-1}=g$.

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References

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