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Gδ space
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Gδ space
In mathematics, particularly topology, a Gδ space is a topological space in which closed sets are in a way ‘separated’ from their complements using only countably many open sets. A Gδ space may thus be regarded as a space satisfying a different kind of separation axiom. In fact normal Gδ spaces are referred to as perfectly normal spaces, and satisfy the strongest of separation axioms.
Gδ spaces are also called perfect spaces. The term perfect is also used, incompatibly, to refer to a space with no isolated points; see Perfect set.
A countable intersection of open sets in a topological space is called a Gδ set. Trivially, every open set is a Gδ set. Dually, a countable union of closed sets is called an Fσ set. Trivially, every closed set is an Fσ set.
A topological space X is called a Gδ space if every closed subset of X is a Gδ set. Dually and equivalently, a Gδ space is a space in which every open set is an Fσ set.
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Gδ space
In mathematics, particularly topology, a Gδ space is a topological space in which closed sets are in a way ‘separated’ from their complements using only countably many open sets. A Gδ space may thus be regarded as a space satisfying a different kind of separation axiom. In fact normal Gδ spaces are referred to as perfectly normal spaces, and satisfy the strongest of separation axioms.
Gδ spaces are also called perfect spaces. The term perfect is also used, incompatibly, to refer to a space with no isolated points; see Perfect set.
A countable intersection of open sets in a topological space is called a Gδ set. Trivially, every open set is a Gδ set. Dually, a countable union of closed sets is called an Fσ set. Trivially, every closed set is an Fσ set.
A topological space X is called a Gδ space if every closed subset of X is a Gδ set. Dually and equivalently, a Gδ space is a space in which every open set is an Fσ set.