nLab binormal topological space

Let $I = [0.1]$ be the closed unit interval with the standard topology. A topological space is binormal if $X\times I$ is a normal topological space (satisfies separation axioms $T_4$ and $T_1$).

Exercise I.B2 in Spanier: If $X$ is a binormal space, $Y$ a normal space and $f:X\to Y$ continuous, the mapping cylinder of $f$ is a normal space.

Borsuk’s homotopy extension theorem. (Exercise I.D2 in Spanier) Let $A$ be a closed subspace of a binormal space $X$. Then $(X,A)$ has the homotopy extension property with respect to any absolute neighborhood retract $Y$.

• Edwin H. Spanier, Algebraic topology, Springer 1966

category: topology