Unformatted text preview: MTH536 Chapter 8
Gene Quinn – p. 1/ Chapter 8
If ¤ is a collection of topologies for a set
§ ¨¦ ¢ £¡ , then is a topology for . ¥ ¢ ¢ £¡ ¥ – p. 2/ Proof of Proposition
For criteria:
§¦ to be a topology for
¥ ¦ , it must satisfy the following three ¢¡ © ¦ ¥ © $¦ ! " # % " # "& ! ¦ ¦ – p. 3/ Proof of Proposition i
Since every is a topology, by deﬁnition and therefore to
¢ '¡ § ¨¦ ¢ ¢ '¡ and belong to each ¥ ¢¡ – p. 4/ Proof of Proposition ii
Suppose
( ( $ 0 1) ¢ £¡ Then since each is a topology for
© ,
$ ) ¢¡ which implies that
© ¢ ¢ '¡ § ¦ ¢ '¡ ¥ – p. 5/ Proof of Proposition iii
Suppose " # ¤ is a an arbitrary collection of open sets in Then
" # $ % ¢ '¡ §¦ . which by the deﬁnition of a topology implies that
" 2 $ ¢ 3¡ ) and therefore
" # " ¢ ¢ '¡ § ¦ " ) ¢¡ © – p. 6/ ...
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 Spring '08
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 Topology, Metric space, Open set, Euclidean space, Topological space

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