Tietze's extension theorem and Urysohn's lemma are closely related in the field of topology. They are both equivalent statements that characterize normal topological spaces. Tietze's extension theorem states that any continuous function defined on a closed subset of a normal space can be extended to a continuous function on the whole space. Urysohn's lemma, on the other hand, asserts that in a normal space, for any two disjoint closed sets, one can find a continuous function that separates them.
The relationship between these two theorems is such that the Tietze extension theorem implies Urysohn's lemma. This implication is demonstrated in various mathematical discussions and proofs, such as in the Stack Exchange post here. Furthermore, both theorems are equivalent to the normality of a topological space, as mentioned in the Wikipedia article on the Tietze extension theorem. Alternative proofs for both theorems using the Cantor function are also presented in a research paper available here.
In summary, Tietze's extension theorem and Urysohn's lemma are deeply interconnected, providing fundamental insights into the properties of normal spaces and the behavior of continuous functions within them.