Let $ \mathscr{T} $ be the category of [insert technical conditions here] topological spaces. Equip $ \mathscr{T}$ with the quillen model structure. The category of based spaces $ \mathscr{T}_*$ inherits a model structure.

**Question 1:**Let $ f,g : X \to Y $ be continuous maps in $ \mathscr{T}$. Is it true that $f$ and $g$ are left homotopic in a model category sense iff $f$ and $g$ are homotopic in an elementary sense?**Question 2:**Let $f,g : X \to Y $ be based maps in $ \mathscr{T}_*$ between cofibrant objects. Is it true that $f $ and $g$ are left homotopic in a model category sense iff $ f $ and $g $ are pointed homotopic in an elementary sense?