Sets can be classified according to the properties they have.

Relative to set theory

• Empty set
• Finite set, Infinite set
• Countable set, Uncountable set
• Power set

Relative to a topology

• Closed set
• Open set
• Clopen set
• F_σ set
• G_δ set
• Compact set
• Relatively compact set
• Regular open set, regular closed set
• Connected set
• Perfect set
• Meagre set
• Nowhere dense set

Relative to a metric

• Bounded set
• Totally bounded set

Relative to measurability

• Borel set
• Baire set
• Measurable set, Non-measurable set
• Universally measurable set

Relative to a measure

• Negligible set
• Null set
• Haar null set

In a linear space

• Convex set
• Balanced set, Absolutely convex set

Relative to the real/complex numbers

• Fractal set

Ways of defining sets/Relation to descriptive set theory

• Recursive set
• Recursively enumerable set
• Arithmetical set
• Diophantine set
• Hyperarithmetical set
• Analytical set
• Analytic set, Coanalytic set
• Suslin set
• Projective set
• Inhabited set

More general objects still called sets

• Multiset

 
icarus set

See also

• List of set identities and relations – Equalities for combinations of sets
• List of types of functions – List of functions in mathematics