Monotonicity and non-satiation

Does Monotonicity o preferences imply their local non-satiation?
I guess yes, because if I draw a small ball around any point, no matter how small the radius is, as long as preferences are monotone, I can always find a point that’s more preferred because it has more of at least one good. Hence local non-satiation.
Is this reasoning correct?

Strict monotonicity implies local nonsatiation (on e.g., R^n).

Monotonicity does not imply local nonsatiation (think of being indifferent between all points). However, the MWG definition of monotone *does* imply local non-satiation (again on e.g., R^n).

The space does actually matter; suppose I have preference on [0,1]^2 represented by u(x,y) = x+y. These preferences are strictly monotone. However, they are not locally non-satiated: (1,1) is a point of satiation.


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