Topics an incoming graduate student should know

Mathematical maturity:

Familiarity with the axiomatic approach, induction, and proof by contradiction.

Set theory:

Subsets; unions and intersections; countability and uncountabilty; de Morgan's laws; Cartesian products; equivalence relations and equivalence classes.

Algebra:

Definition of a group and a normal subgroup; some familiarity with homomorphisms, their kernels and cokernels.

Linear algebra:

Vector spaces, matrices, determinants, linear transformations, eigenvalues, eigenvectors and eigenspaces.

Analysis:

Limits and continuity; lim sup and lim inf; the Riemann integral; epsilon-delta proofs; open, closed, and compact subset of Euclidean space; uniform convergence and uniform continuity; working knowledge of inequalities; partial derivatives and multiple integrals.