Thread: New Question

A subset A of a metric space X is called uniformly discrete if there
epsilon> 0 with the property that the distance between two distinct points of A is always
greater or equal to epsilon:
∀ a, b ∈ A, (a!= b) ⇒ (dX (a, b) ≥ epsilon).
Show that any subset of a uniformly discrete metric space X is
closed in X.

Originally Posted by donsmith

A subset A of a metric space X is called uniformly discrete if there
epsilon> 0 with the property that the distance between two distinct points of A is always
greater or equal to epsilon:
∀ a, b ∈ A, (a!= b) ⇒ (dX (a, b) ≥ epsilon).
Show that any subset of a uniformly discrete metric space X is
closed in X.

Always begin a thread for a new question.

Say tha is fixed by the problem.
If were a limit of some set then the ball must contain a point of distinct from .
But that is impossible.
So is closed by default.

solved