Definition:Singleton
Definition
A singleton is a set that contains exactly one element.
Some authors use the term unit set.
The singleton containing only the element $a$ can be written $\left\{{a}\right\}$.
When this happens, we must be careful to distinguish between the element itself, that is, $a$, and the set containing it, that is, $\left\{{a}\right\}$.
The set $\left\{{a}\right\}$ is known as the singleton of $a$.
Formal Definition
The concept of the singleton set can be formalized rigorously as:
- $\left\{{A}\right\} = \left\{{x : x = A}\right\}$
With this definition, the singleton of proper classes is equal to the empty set.
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Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 3$: Unordered Pairs
- Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.1$: Example $5$
- Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.2$
- George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$
- T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 1$
- Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.3$
