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 December 28th, 2015, 01:25 PM #1 Newbie   Joined: Oct 2015 From: Eskişehir Posts: 11 Thanks: 0 I need examples about open, closed and neither open nor closed sets B[0,1]={f| f:[0,1]-->R, f is continuous} f,g is in B[0,1] d(f,g)=sup{|f(x)-g(x)|} x is in [0,1] I need 6 examples(2 for each) about open subsets, closed subsets and neither open nor closed subsets with respect to information above.
 December 28th, 2015, 08:08 PM #2 Senior Member   Joined: Aug 2012 Posts: 1,922 Thanks: 534 Hint: Can you draw yourself a picture of a neighborhood of the zero function? The zero function is the function f defined by f(x) = 0 for all x.
 March 8th, 2016, 06:04 AM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,195 Thanks: 872 What definitions of "open" and "closed" are you using? There are several different, but equivalent, definitions and how you show that a set is "open" or "closed" depends upon your definition. One common definition, in a metric space with metric d(x,y), is to first say that a point, p, is an "interior point" of set A if and only if there exist a number, r> 0, such that the neighborhood of p, of radius r, {x| d(x,p)< r}, is contained in A. Similarly, we can say that a point, p, is an "exterior point" of set A if and only if there exist a number, r> 0, such that the neighborhood of p, with radius A is contained in the complement of A. Finally, a point, p, is a "boundary point" of set A if and only if it is neither an interior point nor an exterior point of A. Now, a set is "open" if and only if it contains none of its boundary points. A set is closed if an only if it contains all of its boundary points. In the set of real numbers, with the "usual metric", i.e. d(x, y)= |x- y|, a neighborhood of point p, with radius r, is the set of points, x, such that |x- p|< r. A set of the form (a, b), the "open interval" of numbers strictly between a and b, a< x< b, is open because it is easy to see that the "boundary points" are a and b themselves and neither is in the set. It contains neither of its boundary points so is open. Similarly, the "closed interval", [a, b], $\displaystyle a\le x\le b$ also has a and b as boundary points but now includes both of them so is closed. Any easy example of a set that is neither open nor closed is [a, b). Again, its boundary points are a and b. Now it contains a, so is not open, but does not contain b so is not closed. There even exist sets that are [b]both[b] "open" and "closed". Think about that for a moment- to be open a set must contain none of its boundary points. To be closed a set must contain all of its boundary points. What must be true for a set to contain "all" and "none" or its boundary points?
March 8th, 2016, 09:45 AM   #4
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Quote:
 Originally Posted by yuceelly B[0,1]={f| f:[0,1]-->R, f is continuous} f,g is in B[0,1] d(f,g)=sup{|f(x)-g(x)|} x is in [0,1] I need 6 examples(2 for each) about open subsets, closed subsets and neither open nor closed subsets with respect to information above.
Gosh 3x2 = 6 examples, 2 endpoints in the domain of definition.

what a giveaway.

At least you should have got some of these and posted them , which ones are you missing?

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