Topological Groups to Properties and Solutions

[tomboi03]

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?

Thanks

 

[eok20]

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?