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  3. 123 given the following proposition 1215 basic properties of open...

Question: 123 given the following proposition 1215 basic properties of open...

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1.2.3 Given the following Proposition 1.2.15 (Basic properties of open and closed sets). Let (X, d) be a metric space. Prove the following (f) If E1,.... En are a finite collection of open sets in X, then E1n En.n En is also open. If F,.., Fn is a finite collection of closed sets in X, then Fi U Fo U...U Fn is also closed. (g) If EaJael is a collection of open sets in X (where the inder set I could be finite, countable, or uncountable), then the union Uael Ea freX E Ea for some a EIis also open. If FalaEl is a collection of closed sets in X, then the intersection noel FoEX E Fa for all a EI is also closed. (h) If E is any subset of X, then int(E) is the largest open set which is contained in E; in other words, int (E) is open, and given any other open set V C E, we have V C int(E). Similarly E is the smallest closed set which contains E; in other words, E is closed and given any other closed set K E. K > E

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