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 Fσ set Gδ 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
See also List of set identities and relations – Equalities for combinations of sets List of types of functions