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.
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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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#25 part a
![1:12
Aa 4»
Section 3.4 • Topology of the Real Numbers 143
(b) Prove that S is the smallest closed set containing S. That is, show that
SES, 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 that
S° 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 is
open 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 that
Sis 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, is
bounded above, let 6, = sup B; otherwise, let 6, = o. Show that a, eS
and 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,:y<x and y >x.)
(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 SETS
In Section 3.4 we introduced several important topological concepts in R.
Some of these concepts related to points: interior points, boundary points, and
accumulation points. Others related to sets: open sets and closed sets. In this
section 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 will
have a number of special properties not possessed by sets in general. The
first such property is called compactness, and although its definition may at
first appear strange, it is really a widely used concept of analysis. (For
example, see Theorems 5.3.2, 5.3,10, 5.4.6, and 5.5.9.)
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Transcribed Image Text:1:12
Aa 4»
Section 3.4 • Topology of the Real Numbers 143
(b) Prove that S is the smallest closed set containing S. That is, show that
SES, 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 that
S° 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 is
open 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 that
Sis 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, is
bounded above, let 6, = sup B; otherwise, let 6, = o. Show that a, eS
and 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,:y<x and y >x.)
(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 SETS
In Section 3.4 we introduced several important topological concepts in R.
Some of these concepts related to points: interior points, boundary points, and
accumulation points. Others related to sets: open sets and closed sets. In this
section 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 will
have a number of special properties not possessed by sets in general. The
first such property is called compactness, and although its definition may at
first appear strange, it is really a widely used concept of analysis. (For
example, see Theorems 5.3.2, 5.3,10, 5.4.6, and 5.5.9.)
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