I've studied group theory upto isomorphism.
Topics include : Lagrange's theorem, Normal subgroups, Quotient groups, Isomorphism theorems.
I too have done metric spaces and real analysis properly. Can you recommend any good topic to be presented in a short discussion. A good proof on an interesting problem will be highly appreciated.(E.g.- Any subgroup of (R,+) is either cyclic or dense).Is there any such problem which relates number theory and metric spaces or real analysis?
Thanks in advance.
Best Answer
If you have covered elementary point set topology a possibility might be to discuss basics of topological groups. For example, show how having a (continuous) group structure on a topological space simplifies the coarsest separation axioms ($T_0$ implies Hausdorff). Not a cool theorem, but may be the first encounter with homogeneity to some of your audience.
If you want to discuss number theory and metrics, then I would consider Kronecker approximation theorem. Time permitting include the IMHO cool application: given any finite string of decimals, such as $31415926535$, there is an integer exponent $n$ such that the decimal expansion of $2^n$ begins with that string of digits $$ 2^n=31415926535.........? $$ The downside of that is that metric properties take a back seat. You only need the absolute value on the real line and the pigeon hole principle.