Reference

The Tietze extension theorem holds significant historical importance in the field of topology and real analysis. It was first published by Heinrich Tietze in 1915, generalizing Lebesgue's result from the plane to general metric spaces. This theorem allows for the extension of a continuous real-valued function defined on a closed subset of a normal topological space to the entire space, preserving continuity. It has been foundational in the development of general topology and has had a profound impact on mathematical analysis.

The theorem has numerous applications, such as in proving the normality of adjunction of two normal spaces and in real analysis, where it is used to produce sequences of continuous functions that approximate a given function almost everywhere. It is also a key component in proofs involving Urysohn's lemma and paracompactness. The theorem's significance is further highlighted by its role in the development of the concept of subdivisions of cell complexes and its influence on subsequent research and theorems in topology, such as Dugundji's extension theorem.

For more detailed information, you can refer to the Wikipedia page on the Tietze extension theorem or explore the MacTutor History of Mathematics archive for a biography of Heinrich Tietze.