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The union of any number of open sets, or infinitely many open sets, is open. [4] The intersection of a finite number of open sets is open. [4] A complement of an open set (relative to the space that the topology is defined on) is called a closed set. A set may be both open and closed (a clopen set).
In the mathematical field of topology, a G δ set is a subset of a topological space that is a countable intersection of open sets. The notation originated from the German nouns Gebiet ' open set ' and Durchschnitt ' intersection '. [1] Historically G δ sets were also called inner limiting sets, [2] but that terminology is not
A countable intersection of open sets in a topological space is called a G δ set. Trivially, every open set is a G δ set. Dually, a countable union of closed sets is called an F σ set. Trivially, every closed set is an F σ set. A topological space X is called a G δ space [2] if every closed subset of X is a G δ set. Dually and ...
The Baire category theorem (BCT) is an important result in general topology and functional analysis.The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the intersection of countably many dense open sets is still dense).
The Zariski topology on the spectrum of a ring has a base consisting of open sets that have specific useful properties. For the usual base for this topology, every finite intersection of basic open sets is a basic open set. The Zariski topology of is the topology that has the algebraic sets as closed sets.
A good cover is an open cover in which all sets and all non-empty intersections of finitely-many sets are contractible (Petersen 2006). The concept was introduced by André Weil in 1952 for differentiable manifolds , demanding the U α 1 … α n {\displaystyle U_{\alpha _{1}\ldots \alpha _{n}}} to be differentiably contractible.
A G δ set or inner limiting set is a countable intersection of open sets. [8] G δ space A space in which every closed set is a G δ set. [8] Generic point A generic point for a closed set is a point for which the closed set is the closure of the singleton set containing that point. [11]
It is an axiom of topology that the intersection of every finite family of open sets is open; in Alexandrov topologies the finite qualifier is dropped. A set together with an Alexandrov topology is known as an Alexandrov-discrete space or finitely generated space. Alexandrov topologies are uniquely determined by their specialization preorders.