Morse theory is generally presented in the DIFF category. However, there is a version of Morse theory in TOP (see the post *Morse theory in TOP and PL categories?* for references).

It is well known that a TOP (resp. DIFF) closed $n$-manifold that admits a TOP (resp. DIFF) Morse function having exactly two non-degenerate critical points must be homeomorphic to the $n$-sphere.

In DIFF this is due to G. Reeb, and it remains true even if the critical points are degenerate as proven by Milnor and Rosen (Theorem 1, p.124 of [Mil2007]).

In TOP it is due to Kuiper, [Kui1961]; and my first question is: **Does the result remain true even if the points are allowed to be degenerate?**

Working in DIFF, let $M$ be an $n$-manifold and $f$ a Morse function on $M$. Then if $[a,b]$ is an interval of regular values of $f$, it is not difficult to show that $f^{-1}(a)$ is homeomorphic to $f^{-1}(b)$ (e.g. Milnor's *Morse Theory*, p. 12). However, **is this true in the topological case?**

[Kui1961] N.H. Kuiper, A continuous function with two critical points, Bull. Amer. Math. Soc., 67(1961), 281-285.

[Mil2007] J. Milnor, Collected Papers of John Milnor: Vol. III, Differential Topology, AMS, 2007.