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mcastillo356
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- I have some clues, but not a definitive answer to the title question
Hi, PF
is a neighborhood of because there exists an open set such that ; in this case, it might be , which is an open set, for it is an open interval, and provided it satisfies
is not an interval since there are numbers in with real numbers between them that do not belong to , ie, but we see this fact:
It is some neigborhood because if is a topological space...
Formally we say topological space to the ordered pair formed by a set and a topology over , ie, a collection of subsets of that meet all three of the following properties:
1. The empty set and belong to .
2. The intersection of any finite subcollection of sets of is in .
3. The union of any subcollection of sets of is in .
Sets belonging to the topology are called open sets, or simply opens of .
If is a topological space and is a point belonging to , a neighborhood of is a set in which is contained an open set that has as its element the point .
Well, we are in the real line. Why it is not an interval?
An open interval, like , for example, is an open; therefore is a neighborhood of each of its points, not only of . This is used to say that is an interval but I don't think it works to justify it is a neighborhood. For example, let's say
then between the two extremes there is no missing number, that's why it's an interval: however, it is not a neghborhood of neither , since there is no that meets nor is there an open that satisfies .
is a neigborhood of since there is an open such that ; in this case, it might be , which is an open, since it is an open interval, and it also meets
Doubt:
is not an interval because there are reals in with numbers between them that do not belong to , for example, but we have ?
Greetings
Marcos
It is some neigborhood because if
Formally we say topological space to the ordered pair
1. The empty set and
2. The intersection of any finite subcollection of sets of
3. The union of any subcollection of sets of
Sets belonging to the topology
If
Well, we are in the real line. Why it is not an interval?
An open interval, like
then between the two extremes there is no missing number, that's why it's an interval: however, it is not a neghborhood of
Doubt:
Greetings
Marcos
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