(1)Given two locally compact groups G_1, G_2,
if G_1 and G_2 are homeomorphic and algebraically isomorphic,
are they necessarily topologically isomorphic?

(2)Given two Banach spaces X, Y,

if X is isometically isomorphic to a subspace of Y and
Y is isometically isomorphic to a subspace of X,
are they necessarily topologically(or even isometrically) isomorphic?
Remark: It is true if X,Y are Hilbert spaces.

(3)Given two algebraic objects (such as semigroups, groups, rings, field, vector spaces, modules, algebras) X, Y,

if X is monomorphic to Y (ie. there exists a 1-1, homomorphism mapping from X to Y)
and Y is monomorphic to X,
are they necessarily isomorphic?
Remark: It is true if X,Y are vector spaces, so do fields.

Last edited by AHhei on Mon May 09, 2005 7:49 pm; edited 1 time in total