Math Quiz

1. The second order peano axioms under full semantics only model the natural numbers.

2. If there exists a contraction mapping on a compact set X, then X is contractible (null-homotopic).

3. The even and odd numbers form an adjoint modality on the integers as a poset.

4. The rational numbers can be expressed as the product of two abelian groups G and H.

5. The Heine-Borel property (closed and bounded implies compact) holds for the space of continuous functions with the supremum metric.

6. The Peano Axioms only model the natural numbers.

7. Convergent sequences have unique limits if and only if a topological space is Hausdorff.

8. If a function algebra separates points (for every x and y there exists f in A such that f(x) != f(y)) and vanishes nowhere (for all x there exists f in A such that f(x) != 0)

9. There exists a subset X of the reals such that in every interval I, the measure of the intersection of I and X is exactly half of the measure of I (in other words, if you choose a real number at random you have a 50% chance of choosing a point in X).

10. There exists a product and coproduct in the category of fields.

11. The direction field of the equation dy/dx = F(x,y) is quasi-homogenous if and only if the right hand side is quasi-homogeneous and its quasi-homogeneous degree is equal to the difference of the degrees of y and x.

12. Linear Algebra Done Right by Sheldon Axler is a good book.

13. A CW-complex is connected if and only if it is path connected.

14. If S is an integral ring over the image of a ring homomorphism R -> S then the Krull dimension of M as an S-module is the same as the Krull simension of M as an R-module.

15. There exists a zero object in the category of groups.

16. In the category of abelian groups, the product and coproduct are the same object (up to isomorphism)