What is the Usual Topology on R and is R Discrete With Respect to This Topology?
In mathematics, the usual topology on the real numbers R is generated by the open intervals. A set U ⊆ R is open in the usual topology if and only if for every x ∈ U, there exists an open interval (a, b) such that x ∈ (a, b) ⊆ U.
The usual topology on R is not discrete. A topological space is discrete if and only if every subset of the space is both open and closed. However, in the usual topology on R, the set {0} is closed but not open, and the set R is open but not closed. Therefore, R is not discrete with respect to the usual topology.
Related Questions
- What is a topology?
- A topology on a set X is a collection of subsets of X that satisfy certain axioms.
- What is the open interval topology?
- The open interval topology on R is the topology generated by the open intervals.
- What is a discrete topological space?
- A topological space is discrete if and only if every subset of the space is both open and closed.
- Is R a discrete topological space with respect to the usual topology?
- No, R is not discrete with respect to the usual topology.
- Why is R not discrete with respect to the usual topology?
- Because the set {0} is closed but not open, and the set R is open but not closed.
Related Hot Sale Products
- Wilson Tennis Racquet
- Dunlop Badminton Racquet
- Yonex Tennis Bag
- Babolat Badminton Strings
- Head Tennis Shoes
Pre：What is the average salary of a professional polo player
Next：How do I write a passive aggressive apology letter to someone who you have to but someone who has been rude to you
^