Weakly contractible $X$, but none of the maps $*\to X$ are cofibrations

Question

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!

Answer 1

This question was already answered in the comments, but I don't want it to linger forever on the unanswered queue, so I'm making a CW answer summarizing the comments and adding my own example.

Tyrone points out in the comments that the cofibrant objects in the Quillen model structure on $Top$ are compactly generated normal Hausdorff space with CW homotopy type. The OP wants a weakly contractible space that is not cofibrant. Tyler Lawson points out that the Sierpinski space is not cofibrant (because it's not Hausdorff for one reason) and is weakly equivalent to a contractible space. It's also contractible itself. Tyrone also points out that the Warsaw circle is weakly contractible, compact Hausdorff, but does not have the homotopy type of a CW complex and is not contractible.

Another example, that hasn't been mentioned yet, that might help the OP is the long line, which normal and Hausdorff, is weakly contractible but not contractible, and does not have the homotopy type of a CW complex (so, is not cofibrant in the Quillen model structure).

In fact, there are zillions of examples of weakly contractible spaces that are not cofibrant. Even if you shift from $Top$ to the category of compactly generated weak Hausdorff spaces, you don't avoid these examples, e.g., because of the Warsaw circle example. However, there is one option for modeling the homotopy theory of spaces that avoids these kinds of examples, namely the category of simplicial sets, where every object is cofibrant. I am guessing the OP is not yet an expert in algebraic topology, and might benefit from reading more about simplicial sets, e.g., in chapter 3 of Mark Hovey's book or this nice introduction by Greg Friedman.