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a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A
an open set is an abstract concept generalizing the idea of an open interval in the real line
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Is the set of whole numbers is dense?
No. a set of numbers is dense if you always find another number in the set between any two numbers of the set. Since there is no whole number between 4 and 5 the wholes are not dense. The set of rational numbers (fractions) is dense. for example, we can find a nubmer between 2/3 and 3/4 by averaging them and this number (17/24) is once again a rational number. You can always find tha average of two rational numbers and the result is always a rational number, so the ratonals are dense!
Why is every singleton set in a discrete metric space open?
In a metric space, a set is open if for any element of the set we can find an open ball about it that is contained in the set. Well for the singletons in the discrete space, every other element is said to have a distance away of 1. So we can make a ball about the singleton of radius 1/2 ... this ball just equals that singleton since it contains only that element. So it is contained in the set. Thus the singleton set is open.
Derived Set of a set of Rational Numbers?
The derived set of a set of rational numbers is the set of all limit points of the original set. In other words, it includes all real numbers that can be approached arbitrarily closely by elements of the set. Since the rational numbers are dense in the real numbers, the derived set of a set of rational numbers is the set of all real numbers.
What is a set of points that is endless in both direction called?
An open curve
What does a sideways u mean in math?
It is used in set theory and its meaning depends on the way that it is facing. If the open end is to the right then it indicates that the first set is a subset of the second. If the open end is to the left then it indicates that the first set is a superset of the second (the second is a subset of the first).
