Tietze Extension Theorem
If is a normal topological space and is a continuous map from a closed subset , then there exists a continuous map with for all in .
Moreover, may be chosen such that , i.e., if is bounded, may be chosen to be bounded (with the same bound as ). is called a continuous extension of .
Let , be both closed (or both open) subsets of a topological space such that , and let also be a topological space. If both and are continuous, then is continuous.
Let be a closed subset of . Then is closed since it is the preimage of under the function , which is continuous. Similarly, is closed. Then, their union is also closed, being a finite union of closed sets.