Let $M$ be a compact manifold and $f:M \to \mathbb{R}$ a Morse function. For a real number $s$, define $$M^{\leq s}:=\{x \in M \ : \ f(x) \leq s\}.$$ In what follows, let $s<t$ be real numbers such that $[s,t]$ contains no critical value of $f$. The following theorem is well known from Morse Theory:
Theorem. $M^{\leq s}$ and $M^{\leq t}$ are diffeomorphic. Moreover, the natural inclusion $M^{\leq s} \hookrightarrow M^{\leq t}$ is a homotopy equivalence.
This has the following consequence:
Corollary. The homology groups $H_k(M^{\leq s})$ and $H_k(M^{\leq t})$, taken with coefficients in a field, are isomorphic for any $k \in \mathbb{Z}$. The corresponding isomorphism is induced by the natural inclusion $M^{\leq s} \hookrightarrow M^{\leq t}$.
My question: If we replace the weak ineqaulity "$\leq$" in the definition of the sublevel sets $M^{\leq s}$ and $M^{\leq t}$ by a strict one, "$<$", does the theorem or at least the corollary remain true?