# Topological structure (topology)

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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.