Difference between revisions of "Disjunctive sum"
From Encyclopedia of Mathematics
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| + | $#C+1 = 15 : ~/encyclopedia/old_files/data/D033/D.0303320 Disjunctive sum, | ||
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| + | ''disjunct sum, of topological spaces $ X _ \alpha $, | ||
| + | $ \alpha \in A $'' | ||
| + | The space $ Y = \cup _ {\alpha \in A } Y _ \alpha $, | ||
| + | where each $ Y _ \alpha $ | ||
| + | is a copy of $ X _ \alpha $ | ||
| + | and $ Y _ {\alpha _ {1} } \cap Y _ {\alpha _ {2} } = \emptyset $ | ||
| + | for $ \alpha _ {1} \neq \alpha _ {2} $, | ||
| + | while the topology on $ Y $ | ||
| + | is defined by the condition that a set $ U $ | ||
| + | is open in $ Y $ | ||
| + | if and only if its intersection with each $ Y _ \alpha $ | ||
| + | is open. In other words, each $ Y _ \alpha $ | ||
| + | is open and closed in $ Y $. | ||
====Comments==== | ====Comments==== | ||
| − | The space | + | The space $ Y $ |
| + | is also called the discrete sum or the discrete union of the $ X _ \alpha $. | ||
Latest revision as of 19:36, 5 June 2020
disjunct sum, of topological spaces $ X _ \alpha $,
$ \alpha \in A $
The space $ Y = \cup _ {\alpha \in A } Y _ \alpha $, where each $ Y _ \alpha $ is a copy of $ X _ \alpha $ and $ Y _ {\alpha _ {1} } \cap Y _ {\alpha _ {2} } = \emptyset $ for $ \alpha _ {1} \neq \alpha _ {2} $, while the topology on $ Y $ is defined by the condition that a set $ U $ is open in $ Y $ if and only if its intersection with each $ Y _ \alpha $ is open. In other words, each $ Y _ \alpha $ is open and closed in $ Y $.
Comments
The space $ Y $ is also called the discrete sum or the discrete union of the $ X _ \alpha $.
How to Cite This Entry:
Disjunctive sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disjunctive_sum&oldid=46744
Disjunctive sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disjunctive_sum&oldid=46744
This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article