Namespaces
Variants
Actions

Difference between revisions of "Topological structure (topology)"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX)
m (clarification)
Line 2: Line 2:
 
''open topology, respectively, closed 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 $\mathfrak F$.
+
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.
 
$2_\mathfrak G$, 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.

Revision as of 08:45, 8 August 2014

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.


Comments

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

References

[a1] R. Engelking, "General topology" , Heldermann (1989)
How to Cite This Entry:
Topological structure (topology). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topological_structure_(topology)&oldid=32765
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article