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.

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.

Ring Homomorphisms, Residues and Distinct Maximal Ideals

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? ...continues

Maximal Ideals, Residues and Ring Homomorphisms

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 . NOTE: All question marks are Z, the integers Please see the attached ...continues

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 co ...continues

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. ---

Topology

compact spaces and path connectedness (see attachment). --- • Show that if is a homeomorphism between topological spaces, then X is path connected if and only if Y is path connected. Using open cover definition: 1) is a compact subset? 2) Is a compact subset? ---

Let X=X1 x X2 x...x Xn be the product space of the path connected topological space X1, X2, ..., Xn. Prove that the product space X is also path connected.

Let X=X1 x X2 x...x Xn be the product space of the path connected topological space X1, X2, ..., Xn. Prove that the product space X is also path connected. Please see the attached file for the fully formatted problems.

Prove that a set X with discrete topology is a compact topological space if and only if X is a finite set.

Prove that a set X with discrete topology is a compact topological space if and only if X is a finite set. Please see the attached file for the fully formatted problems.

Product Space and Compactness : Let X and Y be two topological spaces. Show that the product space XxY is compact if and only if X and Y are compact.

Let X and Y be two topological spaces. Show that the product space XxY is compact if and only if X and Y are compact. ---

Product Spaces and Hausdorff Spaces : Let X and Y be two topological spaces. Show that the product space XxY is Hausdorff space if and only if X and Y are Hausdorff spaces.

Let X and Y be two topological spaces. Show that the product space XxY is Hausdorff space if and only if X and Y are Hausdorff spaces. ---