What is an example of a dense subset of R with the cofinite topology?
In the context of topology, a dense subset is a set of points that, intuitively, contains "enough" points to represent the entire space. A subset S of a topological space X is said to be dense in X if the closure of S (denoted by Cl(S)) equals X.
A cofinite topology on a set X is a topology where every subset of X with finite complement is open. In other words, a set in a cofinite topology is closed if and only if it is finite.
An example of a dense subset of R with the cofinite topology is the set of all irrational numbers. The closure of the set of all irrational numbers in the cofinite topology is R, since every finite set of irrational numbers has an infinite complement, and hence is not closed. Therefore, the set of irrational numbers is dense in R with the cofinite topology.
Related Questions:
- What is the definition of a dense subset in topology?
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