finalprac

# finalprac - 18.100B and 18.100C Fall 2011 Final exam review...

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: 18.100B and 18.100C Fall 2011 Final exam review sheet Most of the final will consist of problems from this list. 1. Prove that R 2 is not the union of a countable family of lines. 2. Problem 10 from page 44. 3. Let E ⊂ R be uncountable. (a) Prove that E has uncountably many limit points. (b) Prove that there exists x ∈ R such that E ∩ (-∞ , x ) and E ∩ ( x, ∞ ) are both uncountable. 4. Let X be a metric space and ( K n ) n ∈ N a sequence of nonempty compact subsets of X with K n +1 ⊂ K n for all n ∈ N . Prove that if U is an open set with T ∞ n =1 K n ⊂ U , then there exists N ∈ N such that K N ⊂ U . 5. Problem 8 from page 99. 6. Let f : [0 , 1] × [0 , 1] → R be continuous, and let g : [0 , 1] → R be defined by g ( x ) = max { f ( x, y ): y ∈ [0 , 1] } . Prove that g is continuous. 7. Let K be a nonempty compact metric space with metric d , and suppose f : K → K obeys d ( f ( x ) , f ( y )) < d ( x, y ) for all x, y ∈ K with x 6 = y ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

finalprac - 18.100B and 18.100C Fall 2011 Final exam review...

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

View Full Document
Ask a homework question - tutors are online