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Clopen set

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In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. [1]

20 relations: Boolean algebra (structure), Boundary (topology), Closed set, Complement (set theory), Connected space, Discrete space, Empty set, Intersection (set theory), Interval (mathematics), Locally connected space, Mutual exclusivity, Open set, Portmanteau, Rational number, Real line, Stone's representation theorem for Boolean algebras, Subspace topology, Topological space, Topology, Union (set theory).

Boolean algebra (structure)

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.

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Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set.

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Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

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Complement (set theory)

In set theory, the complement of a set refers to elements not in.

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Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

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Discrete space

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense.

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Empty set

In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

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Intersection (set theory)

In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.

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Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.

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Locally connected space

In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.

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Mutual exclusivity

In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur (be true).

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Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

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Portmanteau

A portmanteau or portmanteau word is a linguistic blend of words,, p. 644 in which parts of multiple words or their phones (sounds) are combined into a new word, as in smog, coined by blending smoke and fog, or motel, from motor and hotel.

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Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

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Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers.

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Stone's representation theorem for Boolean algebras

In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets.

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Subspace topology

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).

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Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

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Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

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Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.

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Clopen, Clopen sets, Clopen subset, Closed and open, Closed-open, Closed-open set, Open-closed set.

References

[1] https://en.wikipedia.org/wiki/Clopen_set

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