I'm lost when it comes to this. A clear explanation would help a lot. Maybe some examples including intersections and unions or whatever. I know the definitions but i more need an explanation. Thanks!!!!
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The concepts of sets being either open or closed (or partially open and closed) is a mathematical way of describing certain quite subtle (often geometrical) characteristics of sets. The reason why it may seem a bit hard to comprehend these concepts may be due to their underlying structure of continuity, limits and infinity as I will try to explain.
CLOSED SETS:
A set is closed if it contains a border that distinguishes it from the surrounding elements. For instance a ball is closed if it contains the surface sphere that defines its border.
Every point on the border of a set is recognized as such because however near you come to the border point you will find both points from within the set and points NOT contained in the set. If you consider the closed ball from before and you examine one point on the surface sphere of that ball you will notice that even at the smallest radius from that border point we may find both points within the ball and points outside of the ball.
This notion is then generalized to regard any shape - not just balls.
OPEN SETS
These are characterized by NOT containing the border. An open ball is when you have the sets of points in a ball excluding the points of the surface sphere. So an open ball is a ball with no surface. This seems rather counter intuitive indeed because one may think that any object must have a border... And indeed this open ball has a border - it is just excluded from the set so the border consists of points outside the open ball just merely touching it.
And this is where limits and the concept of continuity arrives. For how can we have points arbitrarily close to the border without including the point on the border? This is an abstraction and has no counter part in how we conceive the world through our senses. In our daily life's conception some of us may think of borders between meeting objects as a pair of borders - one belonging to each object.
CONTINUITY, LIMITS and INFINITY
But in the abstract world of mathematics we realize how it is possible to define a sequence of numbers with no smallest element in it. Think of the sequence 1/n. Which value for n should we choose to pick out the 1/n that is closest to zero? No definite answer can be given. Unless we allow infinity as an answer but then we will encounter other problems:
We must always be careful when dealing with infinity! The problem about infinity is that it has no definite place on any number line and therefore is not really a number - rather than a concept. Some would claim that we just place infinity beyond the other numbers but how is that possible? Then where is ∞-1 to be found? Or put the other way around, if ∞-1=N<∞ then N is a definite number and we then know that N+1 will just be the next number. Hence N+1<∞ but N+1=∞ so ∞<∞ which is really nonsense. Placing ∞ anywhere on a number line would render it finite!!!
CLOSED SETS:
A set is closed if it contains a border that distinguishes it from the surrounding elements. For instance a ball is closed if it contains the surface sphere that defines its border.
Every point on the border of a set is recognized as such because however near you come to the border point you will find both points from within the set and points NOT contained in the set. If you consider the closed ball from before and you examine one point on the surface sphere of that ball you will notice that even at the smallest radius from that border point we may find both points within the ball and points outside of the ball.
This notion is then generalized to regard any shape - not just balls.
OPEN SETS
These are characterized by NOT containing the border. An open ball is when you have the sets of points in a ball excluding the points of the surface sphere. So an open ball is a ball with no surface. This seems rather counter intuitive indeed because one may think that any object must have a border... And indeed this open ball has a border - it is just excluded from the set so the border consists of points outside the open ball just merely touching it.
And this is where limits and the concept of continuity arrives. For how can we have points arbitrarily close to the border without including the point on the border? This is an abstraction and has no counter part in how we conceive the world through our senses. In our daily life's conception some of us may think of borders between meeting objects as a pair of borders - one belonging to each object.
CONTINUITY, LIMITS and INFINITY
But in the abstract world of mathematics we realize how it is possible to define a sequence of numbers with no smallest element in it. Think of the sequence 1/n. Which value for n should we choose to pick out the 1/n that is closest to zero? No definite answer can be given. Unless we allow infinity as an answer but then we will encounter other problems:
We must always be careful when dealing with infinity! The problem about infinity is that it has no definite place on any number line and therefore is not really a number - rather than a concept. Some would claim that we just place infinity beyond the other numbers but how is that possible? Then where is ∞-1 to be found? Or put the other way around, if ∞-1=N<∞ then N is a definite number and we then know that N+1 will just be the next number. Hence N+1<∞ but N+1=∞ so ∞<∞ which is really nonsense. Placing ∞ anywhere on a number line would render it finite!!!