In mathematics, a subset <math>A</math> of a topological space is said to be dense-in-itself if <math>A</math> contains no isolated points. Every dense-in-itself closed set is perfect. Conversely, every perfect set is dense-in-itself. A simple example of a set which is dense-in-itself but not closed (and hence not a perfect set) is the subset of irrational numbers.

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  • In mathematics, a subset <math>A</math> of a topological space is said to be dense-in-itself if <math>A</math> contains no isolated points. Every dense-in-itself closed set is perfect. Conversely, every perfect set is dense-in-itself. A simple example of a set which is dense-in-itself but not closed (and hence not a perfect set) is the subset of irrational numbers. This set is dense-in-itself because every neighborhood of an irrational number <math>x</math> contains at least one other irrational number <math>y \neq x</math>. On the other hand, this set of irrationals is not closed because every rational number lies in its closure. For similar reasons, the set of rational numbers is also dense-in-itself but not closed.
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  • 6228 (xsd:integer)
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  • Dense in-itself
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  • In mathematics, a subset <math>A</math> of a topological space is said to be dense-in-itself if <math>A</math> contains no isolated points. Every dense-in-itself closed set is perfect. Conversely, every perfect set is dense-in-itself. A simple example of a set which is dense-in-itself but not closed (and hence not a perfect set) is the subset of irrational numbers.
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  • Dense-in-itself
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