Connected topological group: Difference between revisions
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# It is connected as a topological space. | # It is connected as a topological space. | ||
# | # The [[connected component of identity|connected component of the identity element]] equals the whole group. | ||
===Equivalence of definitions=== | ===Equivalence of definitions=== | ||
Definitions (1) and ( | Definitions (1) and (2) are clearly equivalent. | ||
==Facts== | |||
# [[Connected implies no proper open subgroup]] | |||
Revision as of 21:13, 12 January 2012
This article defines a property that can be evaluated for a topological group (usually, a T0 topological group)
View a complete list of such properties
Definition
Symbol-free definition
A topological group is termed connected if it satisfies the following equivalent conditions:
- It is connected as a topological space.
- The connected component of the identity element equals the whole group.
Equivalence of definitions
Definitions (1) and (2) are clearly equivalent.