Topological structure (topology)

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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.


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


[a1] R. Engelking, "General topology" , Heldermann (1989)
How to Cite This Entry:
Topological structure (topology). Encyclopedia of Mathematics. URL:
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article