Namespaces
Variants
Actions

Difference between revisions of "Topological structure (topology)"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
Line 1: Line 1:
 +
{{TEX|done}}
 
''open topology, respectively, closed topology''
 
''open topology, respectively, closed topology''
  
A collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t0931401.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t0931402.png" />, of subsets of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t0931403.png" />, satisfying the following properties: 1) The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t0931404.png" />, as well as the empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t0931405.png" />, are elements of the collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t0931406.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t0931407.png" />.
+
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 $\mathfrak F$.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t0931408.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t0931409.png" />. The intersection, respectively union, of a finite number, and the union, respectively intersection, of any number of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t09314010.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t09314011.png" />, 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.
  
Once a topology, or topological structure, has been introduced or defined on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t09314012.png" />, the set is called a [[Topological space|topological space]], its elements are called points and the elements of the collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t09314013.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t09314014.png" />, are called the open, respectively closed, sets of this topological space.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t09314015.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t09314016.png" /> of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t09314017.png" /> is defined, satisfying property 1 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t09314018.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093140/t09314019.png" />, respectively, then the other collection may be defined by duality as consisting of the complements of the elements of the first collection.
+
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
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article