Difference between revisions of "Topological structure (topology)"
From Encyclopedia of Mathematics
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''open topology, respectively, closed topology'' | ''open topology, respectively, closed topology'' | ||
| − | A collection | + | 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 | + | Once a topology, or topological structure, has been introduced or defined on a set $X$, the set is called a [[Topological space|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 | + | 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. |
Revision as of 08:20, 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=32764
Topological structure (topology). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topological_structure_(topology)&oldid=32764
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article