Do there exists nonhomeomorphic closed sets?

yes,

Step-by-step explanation:

Homeomorphic sets :-

Two sets are homeomorphic if define a function f between both sets which is satisfy the following conditions:

 Function f is a bijection (one-to-one and onto),

 Function f is continuous,

 Inverse function of f is also continuous.

example:

Let given by where is a continuous and bijection fuction but inverse of fuction f is discontinuous.

Hence, set   and are not homeomorphic.