**EDIT**:
 1. *First edit after an interesting answer*.
 2. *$(S,\mathcal{T}_1)$ and $(S,\mathcal{T}_2)$ are homotopy equivalent to the same Quillen cofibrant space*.

Let $S$ be a set with two topologies $\mathcal{T}_1$ and $\mathcal{T}_2$ such that the identity map induces a continuous map $(S,\mathcal{T}_1)\to (S,\mathcal{T}_2)$ which is also a homotopy equivalence. I suppose these two topological spaces $\Delta$-generated. 

> Is it enough to conclude that $\mathcal{T}_1=\mathcal{T}_2$ ?

**Motivation**: I need to prove that some continuous bijection is a homeomorphism. Since the spaces are $\Delta$-generated, it suffices to prove that every continuous map from $[0,1]$ to $(S,\mathcal{T}_2)$ is a continuous map from $[0,1]$ to $(S,\mathcal{T}_1)$. I would like to use in some way the homotopy equivalence.