MIchael L.
asked • 11/19/20Theorem 6.7. A set O is open if and only if it is a (finite or infinite) union of open balls.
Prove that A set O is open if and only if it is a (finite or infinite) union of open balls.
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1 Expert Answer
What is the definition of openness that you gave in class? What you want to prove is one of the equivalent statement of openness in a topological space (X, τ) where τ is the topology on Χ.
MIchael L.
Definition 6.5. Let (X, d) be a metric space. A subset O ⊂ X is called open if for each x ∈ O, there is an open ball B ⊂ O containing x
Report
12/02/20
Nikolaos P.
tutor
Call the ball B that you write above B_x. Then you have that O=union over x in O of the balls B_x and you are done.
Report
12/03/20
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Stanton D.
1) Why would a set be a union? A union is considered an object. Do you mean a space, or perhaps all the points within a space? 2) Perhaps you should give a bit of context on "open balls", do they have holes in their surfaces, or what? If they are tangent hollow spheres, I don't see why a union of two, especially if tangent at one point, would be different than a single sphere in terms of openness. Fee free to enlighten me otherwise!11/20/20