In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one.
I assume you are considering the regular topology defined on the set of all real numbers R, in which the set R and empty set are members and hene R is open. Since the topology is closed under complementation in which the complement of an open set is closed, R as the complement of the empty set, an element of the topology, will also be closed. As the previous people answered it, R is both closed and open in this topology. Is there a doar that is both closed and open?
both open and closed.... every real number is represented by a point on the line which is fundamental principle of analytical geometry...
The set of real numbers, R and the empty set, { } are both closed and open.
I assume you are considering the regular topology defined on the set of all real numbers R, in which the set R and empty set are members and hene R is open. Since the topology is closed under complementation in which the complement of an open set is closed, R as the complement of the empty set, an element of the topology, will also be closed. As the previous people answered it, R is both closed and open in this topology. Is there a doar that is both closed and open?
Definitely yes.
From the first axioms of topology, the universe and empty sets are both open and closed. That is, If R is the universe set of any topology, then it is open and closed.
Every real number is represented by a point on the line and is one of fundamental principles. So, it is open as well as closed.
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