Reference

Tietze's extension theorem can indeed be applied to metric spaces. The theorem, which originally generalized Lebesgue's result from the plane to general metric spaces, has been proven for metric spaces by mathematicians such as Hausdorff in 1919. It states that any real-valued, continuous function defined on a closed subset of a normal topological space (which includes many metric spaces) can be extended to a continuous function on the whole space. This extension property is significant in topology and analysis, allowing for the preservation of continuity across broader domains.

The theorem has been discussed and proven in various sources, including on Stack Exchange, where a simpler proof for the metric case is mentioned, and in academic papers available on Project Euclid. It has also been generalized and discussed in the context of complete separable metric spaces on MathOverflow. The theorem's applicability to metric spaces is further supported by resources like the nLab page and various research papers that explore its implications and extensions in different mathematical contexts.

In summary, Tietze's extension theorem is not only applicable to metric spaces but also plays a crucial role in extending continuous functions while maintaining their continuity, a property that is highly valued in many areas of mathematics. For more detailed information, you can refer to sources like the Stack Exchange discussion or the Wikipedia article.