# Is the following map a cofibration?

Assuming that the diagonal map $$X\rightarrow X\times X$$ is a cofibration. Is it true that the diagonal map $$\Sigma X\rightarrow \Sigma X\times \Sigma X$$ is a cofibration? (Where $$\Sigma X$$ is the reduced suspension of $$X$$.)

By definition $$X$$ is locally equiconnected (LEC) iff the diagonal map $$X \to X \times X$$ is a cofibration (of unbased spaces). It follows that $$X \times I$$ is LEC, which implies that $$\Sigma X$$ is LEC, by the Dyer-Eilenberg adjunction theorem for LEC spaces. See Theorem 2.3 in Lewis' paper.