gluing together continuous functions
Suppose we have a collection of subsets of a topological space , and for each we have a continuous function , where is another topological space. If the functions agree wherever their domains intersect, then we can glue them together in the obvious way to form a new function . The theorems in this entry give some sufficient conditions for to be continuous.
Let and be topological spaces, let be a collection of open subsets (http://planetmath.org/OpenSet) of , and let be a function such that the restriction is continuous for all . Then is continuous.
Note that the theorem for closed subsets requires the collection to be locally finite. To see that this is condition cannot be omitted, notice that any function restricts to a continuous function on each singleton, yet need not be continuous itself.
The two theorems are proved in essentially in the same way, but for the first theorem we need to make use of the fact that the union of a locally finite collection of closed sets is closed.
Proof of Theorem 1. Let be a closed subset of . Then . By continuity, each is closed in . But by assumption each is closed in , so it follows that each is closed in . Thus is the union of a locally finite collection of closed sets, and is therefore closed in , and so closed in . So is continuous. ∎
Proof of Theorem 2. Let be an open subset of . Then . By continuity, each is open in . But by assumption each is open in , so it follows that each is open in . Thus is the union of a collection of open sets, and is therefore open in , and so open in . So is continuous. ∎
|Title||gluing together continuous functions|
|Date of creation||2013-03-22 15:17:20|
|Last modified on||2013-03-22 15:17:20|
|Last modified by||yark (2760)|