Unions and intersections of open sets, Real Analysis II
In this video, we aim to prove a three-part statement about open sets in a metric space. We start by establishing that both the entire metric space and the empty set are always open, ensuring that every metric space has at least two open sets. Next, we demonstrate that any union of open sets, regardless of whether the collection is finite, countably infinite, or uncountably infinite, is also open. Lastly, we show that the intersection of a finite number of open sets remains open, contrasting this with the fact that infinite intersections may not necessarily be open.
(MA 426 Real Analysis II, Lecture 6)
In more detail: we begin by proving that the entire space and the empty set are open, which is straightforward. Then, we move on to proving that unions of open sets are open by considering an element in the union and showing that there exists an epsilon neighborhood around this element that is also contained within the union.
Finally, we tackle the proof that finite intersections of open sets are open by carefully constructing an epsilon neighborhood that works for every open set in the intersection. We conclude by discussing counterexamples where the intersection of an infinite collection of open sets is not open, highlighting the importance of these exceptions in understanding the limits of the theorem.
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