Unformatted text preview: MTH536 Chapter 7 Problem 8
Gene Quinn – p. 1/1 Chapter 7 Problem 8
§ Show that § deﬁned by § ¦ § ¡ ¢ £¤ ¦ ¨ £¤ ¥ ¥ © is a homeomorphism between
£¤ and
£¤ ¦ ¥ ¥ © § . – p. 2/1 Proof of proposition
To be a homeomorphism, is 1:1 1) 2) is onto is continuous 3) 4) is continuous must satisfy the following four conditions: – p. 3/1 Proof that
Let is onto
§ be any real number in the interval
£¤ ! and let Then clearly
¤ # ! $ ¦ "¦ ¥ © – p. 4/1 Proof that
Let is onto
§ be any real number in the interval
£¤ ! and let Then clearly
¤ # "¦ Now
% ' % ! ! & § ! & ! ! % ' % ¦" ! ! ¦ ! & & ! ' ! ' ! $ ¦ ¥ © – p. 4/1 Proof that
Let is onto
§ be any real number in the interval
£¤ ! and let Then clearly
¤ # "¦ Now
% ' % ! ! & § ! & ! ! % ' % ¦" ! ! ¦ ! & & ! ' ' ! $ ¦ ¥ © Since
£¤ !( ¥ was an arbitrary choice, for any such that
§ § ! § ! we can ﬁnd an ( £¤ so is onto. ¦ ¥ © – p. 4/1 Proof that
Let result,
¥ 0 ) ( is 1:1
§ § ) with
and § . From the previous £¤ ¦ ¥ and
) ¦" ¦" ) ) ) – p. 5/1 Proof that
Let result,
¥ 0 ) ( is 1:1
§ § ) with
and § . From the previous £¤ ¦ ¥ and
) ¦" ¦" ) Suppose . Then
) 0 ) ) ¦" ) ¦" ) ) – p. 5/1 Proof that
Let result,
¥ 0 ) ( is 1:1
§ § ) with
and § . From the previous £¤ ¦ ¥ and
) ¦" ¦" ) Suppose . Then
) 0 ) ) ¦" ) ¦" ) ) Since and were arbitrary choices, for any § ) 1 § 2 ) 1 ¥ 1 ) ( so is 1:1. )1 ¥ ¦ § £¤ – p. 5/1 Proof that
4 is continuous
5 6 7§ § Let
¤ $ and be an open interval contained in .
£¤ 3 © ¥ ¥ 5 $ 4 © Consider the preimage of
£ 38 9 ¥ 4 $ 5 under , deﬁned by
¦ ¡§ A 7§ § ¥ 4 ¥ $ 5 B £¤ @ ( 3 ¥ © § . Then £¤ – p. 6/1 Proof that
4 is continuous
5 6 7§ § Let
¤ $ and be an open interval contained in .
£¤ 3 © ¥ ¥ 5 $ 4 © Consider the preimage of
£ 38 9 ¥ 4 $ 5 under , deﬁned by
¦ ¡§ A 7§ § ¥ 4 ¥ $ 5 B £¤ If A§ , then
C D§ 4 ¦ A§ A A ¦" 2 2 4 4 4 C 2 A 4 § ¦ 4 @ ( 3 ¥ © § . Then £¤ ¦" so
4 ¦" 4 ¥ F GE ( ¦ 4 – p. 6/1 Proof that is continuous § By a similar argument, if § 5 $ 2 then
5 H IE 2 ( ¤ $ 5 $ 5 5 ¥ ¦" ¦" 5 – p. 7/1 Proof that is continuous § By a similar argument, if § 5 $ 2 then
5 H IE 4 2 ( ¤ $ 5 $ 5 5 ¥ ¦" Note that
5 5 4 4 ¦" when 5 5 ¦" ¦" ¦" 5 4 ¦" so
£ 38 F GE HE P 4 ¥ 5 ¦" 4 ¦" 5 is an open interval contained in . Since was an arbitrary choice, the preimage of any open interval in is an open interval in .
¥ ¦ § 3 £¤ ¥ © ¥ ¦ § £¤ £¤ § 4 §A § A5 ¤ – p. 7/1 Proof that
3 is continuous
§ Now let be any open subset of as a countable union
£¤ 3 . Then can be represented ¥ of disjoint open intervals 3 Q 6 ¥ ¦ § Q . £¤ Q 3 © 3 – p. 8/1 Proof that
3 is continuous
§ Now let be any open subset of as a countable union
£¤ 3 . Then can be represented ¥ of disjoint open intervals 3 Q 6 R S ¥ ¦ § Q . By the proper ties of functions,
Q£ 3 TU8 3 £¤ Q 3 © Q Q Q£ 3 8 Q 3 By the argument above, is an open interval in so the union is an open subset of .
£¤ ¥ ¦ § £¤ § for each ,
V Since the preimage of an arbitrary open set in , is continuous.
£¤ § ¦ § is an open set in £¤ ¥ © ¥ ¦ – p. 8/1 Proof that
Deﬁne ¦ is continuous
§ § by
X W § ¦" X £¤ £¤ ¡W ¥ © ¨ ¥ ¦ 4 5 6 7§ § X Let
¤ $ and be an open interval contained in .
£¤ 3 © ¥ ¥ 5 $ 4 © Consider the preimage of
£ 38 9 ¥ 4 $ 5 under , deﬁned by
¦ ¡§ A 7§ § ¥ 4 ¥ $ 5 B £¤ @ ( 3 ¥ © § . Then £¤ – p. 9/1 ¤ Let
3 5 4 5 6 7§ £¤ ¥ ¥ ¡W £¤ © § ¥ ¨ £¤ 4 ¥ $ © Proof that Deﬁne $ Then for any
X ( ¦ and W X © § § 9 @ ( Y & Y ' ' Y & ' X % Y & Y `% 4 ' ¥ § $ 5 B A 7§ Y `% £¤ ¥ ¦ ¡§ ¥ £¤ 38 $ 5 £ 4 © § ¥ ¦ , Consider the preimage of
§ § ¦ Y & % by is continuous 3 W X § ¦" X X be an open interval contained in . under , deﬁned by
£¤ ¥ © § . Then – p. 9/1 Proof that ¦ is continuous
( ¥ ¦ § and for any ,
' a b% a a a% ' £¤ § § W ' % % ¦ a a so by deﬁnition
W a ' – p. 10/1 Proof that
Let
¤ # 3 $ 4 ¦ is continuous
5 6 7§ § 4 be any open interval contained in , and the preimage of under is
£¤ ¦ £¤ ¥ 3 £ 38 [email protected] ( ¦ ¡§ X A§ W X § W £¤ W 4 ¥ ¥ W $ ¥ 5 $ ¦ 9 5 B ¥ ¦ § . Then – p. 11/1 ¤ Let But
# 4 $ 5 $ ¥ 9 W 4 £ X 2 A ¦ [email protected] ( 38 ¦ Proof that and
3 4 5 6 7§ W X § A 7§ $ 5 2 X $ ¦ 5 5 2 X ( ¤ c ¥ ¦ 5 4 5 H IE F $ E 5 B W ¦ 4 ¥ X § 4 X 2 ( W X A§ 4 4 ¥ £¤ ¦ ¡§ X W £¤ ¥ ¦ § 3 ¦ is continuous
W £¤ ¥ ¦ § be any open interval contained in , and the preimage of under is . Then – p. 11/1 Proof that
Now ¦ is continuous
4 5 £ 38 F dE HE 9 P ¥ ¦ is an open interval in . Since was an arbitrary choice, by an argument similar to the one used to establish the continuity of , we have that is continuous.
¥ © 3 £¤ W § ¦ 4 5 – p. 12/1 ...
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 Spring '08
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 Topology, Metric space, Open set, open interval

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