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Disjunctive sum

From Encyclopedia of Mathematics
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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

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
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