5 Topology Questions (Including: de Morgan's Laws)

1. Prove the following de Morgan's laws:
(a) ...
(b) ...
2. Let A be a set. For each p E A, let Gp be a subset of A such that p C Gp C A. Then show that A = Up E A Gp.
3. Let f : X ---> Y be a function and A, B C Y. Then show that
(a)...
4. Let f : X ?> Y be a function and A C X, B C V. Then show that
(a) A C f-1 o f(A).
(b) B = f o f-1(B).
5. Let f X ?> and g : V ?> Z. Prove that
(a) if f and g are onto, then y o f : X ?'--* Z is onto.
(b) if f and g are one-to-one, then g o f is one-to-one.

Please see the attached file for the fully formatted problems.

Please see the attached file for the complete solution.
Thanks for using BrainMass.

1. Proof:
a. For any , we have , then for any . This implies that for any . So . On the other hand, if , then for any . So for any . This implies that . Thus . Therefore, .
b. For any , we have , then for some . This implies that for some . So . On the other hand, if we have , then for some . This means that for some . So . This implies that . ...

Solution Summary

Topology problems are solved. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

1.
a) Suppose T_1 is a topology on X = {a,b,c} containing {a}, {b} but not {c}. Write down all the subsets of X which you know are definitely in T_1. Be careful not to name subsets which may or may not be in T_1.
b) Suppose T_2 is a topology on Y = {a,b,c,d,e} containing {a,b}, {b,c}, {c,d} and {d,e}. Write down all the subse

? Let X:={a,b,c} be a set of three elements. A certain topology of X contains (among others) the sets {a}, {b}, and {c}. List all open sets in the topology T.
? Let X':={a,b,c,d,e} be a set of five elements. A certain topology T' on X' contains (among others) the sets {a,b,c}, {c,d} and {e}. List any other open set in T' which

Suppose (X,T) is a topological space. Let Y be non-empty subset of X. The the set J={intersection(Y,U) : U is in T} is called the subspace toplogy on Y. Prove that J indeed a toplogy on Y i.e., (Y,J) is a topological space.

Can anyone show me a simple definition of De Morgan's Law (not proof just a definition using union and intersection symbols)?
Please show a simple example of the use of De Morgan's Law.

I have these problems from Topology of Surfaces by L.Christine Kinsey: the problems I require assistance with are 2.26, 2.28, 2.29, and 2.32. These are stated below.
PROBLEM (Exercise 2.26). Describe what stereographic projection does to
(1) the equator,
(2) a longitudinal line through the north and south poles,
(3) a tr