An Effective Tietze-Urysohn Theorem for QCB-Spaces

The Tietze-Urysohn Theorem states that every continuous real-valued func- tion defined on a closed subspace of a normal space can be extended to a continuous function on the whole space. We prove an effective version of this theorem in the Type Two Model of Effectivity (TTE). Moreover, we introduce for qcb-spaces a slightly weaker notion of normality than the classical one and show that this property suffices to establish an Extension Theorem for continuous functions defined on functionally closed subspaces. Qcb-spaces are known to form an important subcategory of the cat- egory Top of topological spaces. QCB is cartesian closed in contrast to Top.