Disjunctive sum

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 $.