Perfect setIn general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set S {\displaystyle S} is perfect if S = S ′ {\displaystyle S=S'} , where S ′ {\displaystyle S'} denotes the set of all limit points of S {\displaystyle S} , also known as the derived set of S {\displaystyle S} . (Some authors do not consider the empty set to be perfect.)
Polish spaceIn mathematics, a Polish space is a separable, completely metrizabletopological space; i.e., a space homeomorphic to a completemetric space that has a countabledense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians, such as Sierpiński, Kuratowski, Tarski and others.Descriptive set theoryUncountable setIn mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than aleph-null, the cardinality of the natural numbers. Examples of uncountable sets include the set R {\displaystyle \mathbb {R} } of all real numbers and the set of all subsets of the natural numbers.
Countable setMathematical set that can be enumeratedA mathematical set is countable if either it is finite or it can be put in one to one correspondence with the set of natural numbers.
Cardinality of the continuumIn set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbersR{\displaystyle \mathbb {R} }, sometimes called the continuum. It is an infinitecardinal number and is denoted by c{\displaystyle {\mathbf {\mathfrak {c}}}} (lowercase Fraktur "c") or |R|.{\displaystyle {\mathbf {|}}{\mathbf {\mathbb {R} }}{\mathbf {|}}.}The real numbers R{\displaystyle \mathbb {R} } are more numerous than the natural numbersN{\displaystyle \mathbb {N} }.Continuum hypothesisIn mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: There is no set whose cardinality is strictly between that of the integers and the real numbers. The name of the hypothesis comes from the term continuum for the real numbers.
SubsetIn mathematics, a setA is a subset of a set Bif and only if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements.
Empty setIn mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called non-empty.
MathematicsField of knowledgeIlluminated letter P at the beginning of Adelard of Bath's translation of Euclid's Elements, which starts "Punctum est illud cui pars non est" ('A point is that which has no part'). Geometry is shown personified as a woman, following Martianus Capella's De nuptiis Philologiae et Mercurii.Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities.
Real numberNumber representing a continuous quantityReal numbers can be thought of as all points on a number line.In mathematics, a real number is a number that can be used to measure a continuous one-dimensionalquantity such as a length, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.
CounterexampleA counterexample is a specific example that contradicts a claim, hypothesis, or generalization. In logic a counterexample disproves a universally stated claim, and does so rigorously in the fields of mathematics and philosophy. For example, the statement that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universally quantified "all students are lazy."
Polish spaceIn mathematics, a Polish space is a separable, completely metrizabletopological space; i.e., a space homeomorphic to a completemetric space that has a countabledense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians, such as Sierpiński, Kuratowski, Tarski and others.Descriptive set theoryUncountable setIn mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than aleph-null, the cardinality of the natural numbers. Examples of uncountable sets include the set R {\displaystyle \mathbb {R} } of all real numbers and the set of all subsets of the natural numbers.Countable setMathematical set that can be enumeratedA mathematical set is countable if either it is finite or it can be put in one to one correspondence with the set of natural numbers.Cardinality of the continuumIn set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbersR{\displaystyle \mathbb {R} }, sometimes called the continuum. It is an infinitecardinal number and is denoted by c{\displaystyle {\mathbf {\mathfrak {c}}}} (lowercase Fraktur "c") or |R|.{\displaystyle {\mathbf {|}}{\mathbf {\mathbb {R} }}{\mathbf {|}}.}The real numbers R{\displaystyle \mathbb {R} } are more numerous than the natural numbersN{\displaystyle \mathbb {N} }.Continuum hypothesisIn mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: There is no set whose cardinality is strictly between that of the integers and the real numbers. The name of the hypothesis comes from the term continuum for the real numbers.SubsetIn mathematics, a setA is a subset of a set Bif and only if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements.Empty setIn mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called non-empty.MathematicsField of knowledgeIlluminated letter P at the beginning of Adelard of Bath's translation of Euclid's Elements, which starts "Punctum est illud cui pars non est" ('A point is that which has no part'). Geometry is shown personified as a woman, following Martianus Capella's De nuptiis Philologiae et Mercurii.Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities.Real numberNumber representing a continuous quantityReal numbers can be thought of as all points on a number line.In mathematics, a real number is a number that can be used to measure a continuous one-dimensionalquantity such as a length, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.CounterexampleA counterexample is a specific example that contradicts a claim, hypothesis, or generalization. In logic a counterexample disproves a universally stated claim, and does so rigorously in the fields of mathematics and philosophy. For example, the statement that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universally quantified "all students are lazy."*As an Amazon Associate I earn from qualifying purchases.