Connected spaces

If X is a connected space containing more than one point, and if {x} is closed subset for every x is a member of X show that the number of points in X is infinite.

Homomorphisms

(See attached file for full problem description with proper symbols) --- • Let . Prove that the map given by , where is the residue of a modulo n, is a ring homomorphism. Find the kernel and image of . • Prove that if is a ring homomorphism, then given by is also a ring homomorphism. • Write down two distinct maximal ideals of . Does have a finite or infinite number of maximal ideals? Give brief reasons for your answer. ---

Maximal ideals

(See attached file for full problem description with symbols) NOTE: All question marks are Z, the integers --- • Let . Show that the map the residue of a+ b modulo 2, is a ring homomorphism with . Prove that . Hence, or otherwise, give a maximal ideal of . • Consider the ideal (2)+(x) of . Show that (2)+(x) . Hence explain why (x) is not a maximal ideal of . ---

Path components

(See attached file for full problem description with proper symbols and equations) --- Let X be a topological space. Mapping a point to the path component which contains x establishes a map . Show that for any continuous map between topological spaces, there exists a map such that the following holds: • • for two continuous maps and we have • for the identity we have where the latter map denotes the identity on . ---

Path connected problems

(See attached file for full problem description with proper symbols) --- • Show that, for , the sphere is path connected. • Show that if f:X->Y is a continuous map between topological spaces and X is path connected, then the image f(Y) is also path connected. ---