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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: 2,212 Thanks: 655 
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: 3,264 Thanks: 902 
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  
Senior Member Joined: Jun 2015 From: England Posts: 891 Thanks: 269  Quote:
what a giveaway. At least you should have got some of these and posted them , which ones are you missing?  

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