A computer is infected with the Sasser virus.

A computer is infected with the Sasser virus. Assume that it infects 20 other computers within 5 minutes; and that these PCs and servers each infect 20 more machines within another five minutes, etc. How long until 100 million computers are infected?

sierpinski space is contractible

Let X be Sierpinski space: X={x,y} with topology {X,empty set, {x}} . prove that X is contractible.

Path connected subsets

Give proofs or counter-example for the following statements: i)If X and Y are path connected subsets of Z and X/Y ( X intersection Y) is non empty then X/Y is path connected. ii) If X and Y are path connected subsets of Z the X/Y(union) is path-connected. iii)If X and Y are path connected subsets of Z and X/Y ( X intersection Y) is non empty then X/Y(union) is path connected.

Quotient map and open map

Let be p:G to G/N a quotient map.Is it open? Let be f:G to H an open map. Is it quotient map? Here G and H are topological groups, and N is an subgroup.

Definition of degree

Let f: S^n --> S^n be a continuous map. Consider the induced homomorphism f*: H~_n (S^n) --> H~_n (S^n), where H~_n is a reduced homology group. Then from the fact that H~_n (S^n) is an infinite cyclic group, it follows that there is a unique integer d such that f*(u) = du for any u in H~_n (S^n). How exactly does "there is a unique integer d such that..." follow from "H~_n (S^n) is an infinite cyclic group"?