**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 $\mathcal{T}\_1=\mathcal{T}\_2$ ?
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**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.



Continuous bijection between two homotopy equivalent $\Delta$-generated spaces
Let $X = S^1$ and $X^\delta$ its discretisation. Then the identity map $\iota: X^\delta \to X$ is a continuous bijection of $\Delta$-generated spaces, although not a homotopy equivalence. But the map of unreduced cones $$C(\iota): C(X^\delta) \to C(X)$$ is a homotopy equivalence and a continuous bijection, and both spaces are $\Delta$-generated.


