Connected topological group

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Revision as of 21:29, 12 January 2012 by Vipul (talk | contribs) (Alternative definition for a locally connected topological group)
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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:

  1. It is connected as a topological space.
  2. The connected component of the identity element equals the whole group.

Equivalence of definitions

Definitions (1) and (2) are clearly equivalent.

Alternative definition for a locally connected topological group

For a locally connected topological group, being connected is equivalent to having no proper open subgroup.

In particular, this alternate definition applies to algebraic groups equipped with the Zariski topology, as well as to Lie groups. For more, see equivalence of definitions of connected algebraic groups and equivalence of definitions of connected Lie groups.

Facts

  1. Connected implies no proper open subgroup