This is from Munkres, 2nd ed., Ch2, supp. exercise 7(a).

__Definitions__:

__Topological Group__:

is a topological group with identity

. This means that the product function

is continuous and

is a continuous map.

For

subsets of

.

.

__Symmetric neighborhood__: A neighborhood

of

such that

. Note that, for U open in G and

,

and

are both symmetric.

__Claim__: Any neighborhood U of e contains a symmetric neighborhood

such that

.

I can show that if V is symmetric, that

is an open set in

and

trivially. But I'm not seeing how to prove that a construction of a symmetric V from U has

?