Facts about open sets in metric spaces 1 Both 0 and X are op

Facts about open sets in metric spaces (1) Both 0 and X are open in (X, d) (2) Ifu and v are open in d), then unv is open in (X, d) (3) If tur) is a collection of open sets in (X, d) then u Uaua is open. Question 7.6 Prove the above facts about open sets in a metric space. requires much work. The others Note: Showing (2) is the only one of the above three facts that follow fairly directly once you understand the definition of open. For instance, showing e is open, just requires realizing that there nothing in the definition of open is vacuously true. (2) requires that you recognize that if oi 52 then B(ar, 51) g B(r, 52).

Solution

a) The null set is a set of measure zero. In a metric space, it may consist of a set of finite points where distance =0.

In that case null set will be closed. But here since X is an open set null set may not consist of finite number of points with distance zero. Hence null set is an open set. Also X is an open set.

b) Let P = U intersection V.

Consider any x in P then x belongs to both U and V

Since both U and V are open, this implies there is always an open neighbourhood (open ball) around x.

Hence if we assume that P is not open, then there is an x such that x is on the boundary of either U or V.

But since U and V are open this is not possible. Hence our assumption was wrong.

For all x in P there is an open ball around it. IN other words, U int V is always open when U and V are open.

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3) To prove that union of a number of open sets is open.

Take any x in U. This means u must be contained in atleast one specific Ua.

But then that neighborhood of x must also be contained in the union of U.

Hence, any x in U has a neighborhood that is also in U, which means by definition that U is open.