Show that any open interval is an open set. (Note: An open interval has the form (a, b) where a, b ∈ ℝ and a < b. Under this definition, (-∞, a) and (a, ∞) are not open intervals.)
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Let's take an arbitrary point x in (a, b). Since a < x < b, we can choose two positive real numbers r1 and r2 such that a < x - r1 and x + r2 < b. Then, the open ball centered at x with radius r = min(r1, r2) is entirely contained in (a, b). To see why, suppose Show more…
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