Are loop spaces of homotopically equivalent spaces homotopically equivalent? Let $f:X \to Y$ be a homotopy equivalence of pointed topological spaces.
Then, is the induced map of pointed loop spaces $\Omega (f): \Omega X \to \Omega Y$ a homotopy equivalence?
Here, loop spaces are equipped with the compact-open topologies.
Is there any counterexample?
I do not know even whether the induced map $\Omega (f)$ is continuous or not in general.
 A: Is the problem that you want to allow "pathological" topological spaces? I would have thought that for nice spaces all of this were sort of true by definition. (after all, $\Omega(X)=\bullet \times^h_X \bullet$ and homotopy limits are invariant under weak equivalences, no?)
A: The answer is yes always, provided $f\colon\thinspace (X,x_0)\to (Y,y_0)$ is a pointed homotopy equivalence of pointed spaces (meaning that the homotopies $g\circ f \simeq 1_X$ and $f\circ g\simeq 1_Y$ preserve the base points).
This follows from the fact that $\Omega$ is a homotopy functor from based spaces to based spaces, meaning in particular that 
$$\Omega(g)\circ\Omega(f) = \Omega(g\circ f) \simeq \Omega(1_X) = 1_{\Omega X}$$ 
and
$$\Omega(f)\circ\Omega(g) = \Omega(f\circ g) \simeq \Omega(1_Y) = 1_{\Omega Y}.$$
To see that $\Omega$ is a homotopy functor, note that $\Omega(f)$ takes a loop $\gamma\colon\thinspace I\to X$ to the composition $f\circ \gamma\colon\thinspace I\to Y$. So if $F_t\colon  X\to Y$ is a pointed homotopy from $f$ to $f'$, then 
$\Omega(F_t)\colon\thinspace \Omega(X)\to \Omega(Y)$ is a pointed homotopy from $\Omega(f)$ to $\Omega(f')$.
