Let $\mathrm{Top}$ be the category of all topological spaces and continuous maps. The Quillen model structure on $\mathrm{Top}$ has weak equvalences $W = \{ \text{weak homotopy equivalences} \}$, fibrations $F = \{ \text{Serre fibrations} \}$ and cofibrations $C = \{ \text{retracts of relative cell complexes} \}$.

Let $* \in \mathrm{Top}$ be the one-point space. I am interested in a space $X \in \mathrm{Top}$ with the following two properties.

- There is a weak equivalence $* \to X$. (Hence all maps $* \to X$ are weak equivalences.)
- None of the maps $* \to X$ are cofibrations.

Does such a space $X$ exist?

If yes, I am interested in an example, or a family of examples. If not, why not? There must be something about $\mathrm{Top}$ since there exist model categories with such an $X$. Are there any additional conditions on the category that would either guarantee or prevent $X$ from existing?

Thank you!