#25 part a 1:12Aa 4»Section 3.4 • Topology of the Real Numbers 143(b) Prove that S is the smallest closed set containing S. That is, show thatSES, and if Cis any closed set containing S, then SsC.(c) Prove that 5 = cl S.(d) IfS is bounded, prove that S is bounded.25. For any set SCR, let S* denote the union of all the open sets contained in S.(a) Prove that S is an open set.(b) Prove that S° is the largest open set contained in S. That is, show thatS° S S, and if U is any open set contained in S, then USS".(c) Prove that S- int S.26. In this exercise we outline a proof of the following theorem: A subset of R isopen iff it is the union of countably many disjoint open intervals in R.(a) Let S be a nonempty open subset of R. For each x e S, let A, ={a e R: (a, x]C S} and let B, = {b eR: [x, b) CS}. Use the fact thatSis open to show that A, and B, are both nonempty.(b) If A, is bounded below, let a, - inf A. Otherwise, let a, - -0, If B, isbounded above, let 6, = sup B; otherwise, let 6, = o. Show that a, eSand b, e S.(c) Let I, be the open interval (a,, b.). Clearly, x e I, Show that I,S .(Hint: Consider two cases for y e l,:yx.)(d) Show that S=U.s.(e) Show that the intervals ,:x e S} are pairwise disjoint. That is,suppose x, y eS with xy. IfI, nl, 0, show that /,-1,.() Show that the set of distinct intervals {I,:x e S} is countable.Section 3.5 COMPACT SETSIn Section 3.4 we introduced several important topological concepts in R.Some of these concepts related to points: interior points, boundary points, andaccumulation points. Others related to sets: open sets and closed sets. In thissection we define another type of set that occurs frequently in applications.If we require a subset of R to be both closed and bounded, then it willhave a number of special properties not possessed by sets in general. Thefirst such property is called compactness, and although its definition may atfirst appear strange, it is really a widely used concept of analysis. (Forexample, see Theorems 5.3.2, 5.3,10, 5.4.6, and 5.5.9.)143日ReaderContentsNotebookBookmarksFlashcards

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#25 part a 1:12Aa 4»Section 3.4 • Topology of the Real Numbers 143(b) Prove that S is the smallest closed set containing S. That is, show thatSES, and if Cis any closed set containing S, then SsC.(c) Prove that 5 = cl S.(d) IfS is bounded, prove that S is bounded.25. For any set SCR, let S* denote the union of all the open sets contained in S.(a) Prove that S is an open set.(b) Prove that S° is the largest open set contained in S. That is, show thatS° S S, and if U is any open set contained in S, then USS&#34;.(c) Prove that S- int S.26. In this exercise we outline a proof of the following theorem: A subset of R isopen iff it is the union of countably many disjoint open intervals in R.(a) Let S be a nonempty open subset of R. For each x e S, let A, ={a e R: (a, x]C S} and let B, = {b eR: [x, b) CS}. Use the fact thatSis open to show that A, and B, are both nonempty.(b) If A, is bounded below, let a, - inf A. Otherwise, let a, - -0, If B, isbounded above, let 6, = sup B; otherwise, let 6, = o. Show that a, eSand b, e S.(c) Let I, be the open interval (a,, b.). Clearly, x e I, Show that I,S .(Hint: Consider two cases for y e l,:yx.)(d) Show that S=U.s.(e) Show that the intervals ,:x e S} are pairwise disjoint. That is,suppose x, y eS with xy. IfI, nl, 0, show that /,-1,.() Show that the set of distinct intervals {I,:x e S} is countable.Section 3.5 COMPACT SETSIn Section 3.4 we introduced several important topological concepts in R.Some of these concepts related to points: interior points, boundary points, andaccumulation points. Others related to sets: open sets and closed sets. In thissection we define another type of set that occurs frequently in applications.If we require a subset of R to be both closed and bounded, then it willhave a number of special properties not possessed by sets in general. Thefirst such property is called compactness, and although its definition may atfirst appear strange, it is really a widely used concept of analysis. (Forexample, see Theorems 5.3.2, 5.3,10, 5.4.6, and 5.5.9.)<>143日ReaderContentsNotebookBookmarksFlashcards

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25. For any set S CR, let S denote the union of all the open sets contained in S. (a) Prove that S is an open set.