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Topological groups

  1. Mar 24, 2009 #1
    If A and B are subsets of G, let A*B denote the set of all points a*b for a
    in A and b in B. Let A^(-1) denote the set of all points a^(-1), for a in A.

    a)A neighborhood V of the identity element e is said to be symmetric if V = V^(-1)
    . If U is a neighborhood of e, show there is a symmetric neighborhood V of e such that
    V*V/subset of U.[Hint: if W is a neighborhood of e, then W*W^(-1) is symmetric.

    b)Show that G is Hausdorff. In fact, show that if x not equals y, there is a neighborhood
    V of e such that V*x and V*y are disjoint.

    c)Show that G statisfies the following separation axiom, which is called the regularity axiom:
    Given a closed set A and a point x not in A, there exist disjoint open sets containing A and x,
    repectively. [Hint: There is a neighborhood V of e such that V*x and V*A are disjoint.]

    d)let H be s subgroup of G that is closed in the topology of G; let p:G-->G/H be the quotient map.
    Show that G/H satisfies the regularity axiom.[Hint: Examine the proof of (c) when A is saturated.]

    idk how to do any of this... can someone help me out?

  2. jcsd
  3. Mar 25, 2009 #2
    I don't think the hint is necessary for part a. Consider the set U intersect U^{-1}. Can you show that it is open, symmetric, and contains e?
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