Noun interpretation

Definition: A set without any elements is called an empty set. Representation method: use the symbol φ.

The essence of empty set: an empty set is a subset of all sets.

For any set a, an empty set is a subset of a;

? A: {}? A

For any set a, the union of empty set and a is a:

? A: A ∨ {} = A.

For any set a, the intersection of an empty set and a is an empty set: something does not exist, it is an empty set.

? A: A ∩ {} = {}

For any set a, an empty set and the cartesian product of a are empty sets;

? A: A × {} = {}

The only subset of an empty set is the empty set itself:

? A: A? {} ? A = {}

The number of elements in an empty set (that is, its potential) is zero; In particular, the empty set is limited:

|{}| = 0

In set theory, two sets are equal if they have the same elements; Then only one set can have no elements, that is, the empty set is unique.

Considering that an empty set is a subset of a real number line (or any topological space), it is both an open set and a closed set. The boundary point set of an empty set is an empty set and its subset, so an empty set is a closed set. The interior point set of an empty set is also an empty set and a subset of it, so an empty set is an open set. In addition, an empty set is a compact set because all finite sets are compact.

The closure of an empty set is an empty set.

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