Show that points outside a closed set are separated from it.

[code06]

Let (X, d) be a metric space, G ⊆ X be a closed set, and x ∈ X \ G be a point outside G. Show that there exists an ε ∈ (0,∞) such that B_ε (x)∩ G = ∅.

 

[Deleted member 4993]

Let (X, d) be a metric space, G ⊆ X be a closed set, and x ∈ X \ G be a point outside G. Show that there exists an ε ∈ (0,∞) such that B_ε (x)∩ G = ∅.

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[pka]

Let (X, d) be a metric space, G ⊆ X be a closed set, and x ∈ X \ G be a point outside G. Show that there exists an ε ∈ (0,∞) such that B_ε (x)∩ G = ∅.

HINT: The complement of a closed set is an open set.