Are homotopy equivalent path-connected spaces homotopy equivalent as pointed spaces? Let $(X,x)$ and $(Y,y)$ be path-connected pointed topological spaces.
Is it true that the statement
''$X$ and $Y$ are homotopy equivalent''
implies
''$(X,x)$ and $(Y,y)$ are homotopy equivalent as pointed spaces''? How can I prove this?
A pointed homotopy equivalence is a homotopy equivalence where the two involved homotopies are constant on the basepoints at every ''time''.
 A: The claim is false. Here is a standard exercise in algebraic topology. Define the following space $X$:
$$X = \{ (t x, t) \in \mathbb{R}^2 : x \in \mathbb{Q} \cap [0, 1], t \in [0, 1] \}$$
Clearly, $X$ is contractible; however, $(X, (0, 1))$ is not homotopy equivalent to the point as a pointed topological space!
A: Let me add to Zhen Lin's response an affirmative answer in the case of CW complexes. Namely, if $f:(X,x_0)\rightarrow(Y,y_0)$ is a homotopy equivalence (in the non-pointed sense) between CW complexes, where $x_0$, $y_0$ are 0-cells, then it is a homotopy equivalence rel basepoints. In fact, the same statement holds true for any pairs $(X, x_0)$ and $(Y,y_0)$ having the homotopy extension property.
This follows from, for example, Proposition 0.19 of Allen Hatcher's "Algebraic Topology," noting that CW pairs have the homotopy extension property and that the hypothesis that $A$ is a common subspace to $X$ and $Y$ in the proposition is no restriction on its application here.
Let me know if I am doing anything crazy.
