245Chw1sol

# 245Chw1sol - Math 245C Spring 2007 Homework 1 Solutions I...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 245C Spring 2007 Homework 1 Solutions: I read problems 10, 26, 43, 49, and 74. Here are correct solutions to those and to problem 13, which is needed for Problem 49. 10. (a.) If A ⊂ X is open and closed (called “clopen” by some) and ∅ negationslash = A negationslash = X then A and X \ A are nonempty disjoint open sets with union X , so that X is not connected. If X is disconnected, then X = U ∪ V with U ∩ V = ∅ , and U,V both proper. Then A = U is clopen and ∅ negationslash = A negationslash = X . (b.) Assume X = uniontext A E α = U ∪ V where U ∩ V = ∅ , ∅ negationslash = U,V negationslash = X, and U and V are (relatively) open in X . Then for each α , U ∩ E α and V ∩ E α are disjoint relatively open subsets of E α with ( U ∩ E α ) ∪ ( V ∩ E α ) = E α . Since E α is connected it follows that E α = U ∩ E α or E α = V ∩ E α , and since U negationslash = ∅ and V negationslash = ∅ , each alternative holds for at least one α ∈ A. Take α U ∈ A such that E α U ⊂ U, and α V ∈ A such that E α V ⊂ V. Then intersectiontext A E α ⊂ E α U ∩ E α V = ∅ , a contradiction. (c.) Suppose not. Then there are open sets U 1 and U 2 such that A = ⊂ ( U 1 ∪ U 2 ), A ∩ U i negationslash = ∅ for i = 1 , 2, and A ∩ U 1 ∩ U 2 = ∅ . By the definition of closure, A ∩ U i negationslash = ∅ for i = 1 , 2, and by the above A ⊂ ( U 1 ∩ U 2 ), and A ∩ U 1...
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

245Chw1sol - Math 245C Spring 2007 Homework 1 Solutions I...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online