{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

homework6mat25

# homework6mat25 - metric Deﬁne d x,y = min 1,d x,y Verify...

This preview shows page 1. Sign up to view the full content.

Homework 6 due Wednesday September 10th 1. Consider the set S = [0 , 3) (3 , 5). What is the interior of S ? What are the limit points of S ? What is the closure of S ? 2. If A is open and B is closed, prove that A \ B is open. 3. Let S be a bounded inﬁnite set and let x = sup S . Prove that either x S or x is a limit point of S . 4. Use the deﬁnition of compactness to prove that [1 , 3) is not compact. 5. Let S = R × R and let d ( x,y ) = p ( x 1 - y 1 ) 2 + ( x 2 - y 2 ) 2 be the Euclidean
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: metric. Deﬁne d * ( x,y ) = min { 1 ,d ( x,y ) } . Verify that d * is a metric on S . Draw neighborhoods around the origin of radius 1 / 2, 1, and 2. 6. If A and B are compact subsets of a metric space ( X,d ), prove that A ∪ B is also compact. 7. Let x be a point in the metric space ( X,d ). Prove that the singleton set { x } is closed. 1...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online