Topology - Would you be able to give me an concrete example or problem that fits this theorem? And give a step by step proof so I may understand the logic. And please explain the theorem. Thank you.

THEOREM 3.1. Let (X,U) be the subspace [0, 1] of the real axis, and let =F(U) be the induced fuzzy topology for U. Then (X, F(U)) is C, but is not C. THEOREM 3.1. Let (X,U) be the subspace [0, 1] of the real axis, and let =F(U) be the induced fuzzy topology for U. Then (X, F(U)) is C, but is not C