We say that a Topological space $Y$ satisfies the Tietze extension propery, TE property, if in the formulation of Tietze extension theorem $Y$ can be replaced by "$\mathbb{R}$".

Obvioysly the product of two TE spaces is again a TE space. But what about a twist product? More precisely assume that the fiber and base space of a fiber bundle satisfy TE property. Is the total space necessaryly a TE space?