Topological structure (topology)

open topology, respectively, closed topology

A collection $\mathfrak G$, respectively $\mathfrak F$, of subsets of a set $X$, satisfying the following properties:

$1$. The set $X$, as well as the empty set $\emptyset$, are elements of the collection $\mathfrak G$, respectively $\mathfrak F$.

$2_\mathfrak G$, respectively $2_\mathfrak F$. The intersection, respectively union, of a finite number, and the union, respectively intersection, of any number of elements of $\mathfrak G$, respectively $\mathfrak F$, is an element of the same collection.

Once a topology, or topological structure, has been introduced or defined on a set $X$, the set is called a topological space, its elements are called points and the elements of the collection $\mathfrak G$, respectively $\mathfrak F$, are called the open, respectively closed, sets of this topological space.

If one of the collections $\mathfrak G$ or $\mathfrak F$ of subsets of $X$ is defined, satisfying property 1 and $2_\mathfrak G$ or $2_\mathfrak F$, respectively, then the other collection may be defined by duality as consisting of the complements of the elements of the first collection.

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See also Topology, general; Topological space; General topology.

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