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I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand Example 3.10 (b) on page 95 ... ... Example 3.10 (b) reads as follows:
View attachment 9208
My question is as follows:Stromberg says that if \(\displaystyle X\) is any set and \(\displaystyle \mathscr{T}\) is the family of all subsets of \(\displaystyle X\) ...
... then \(\displaystyle \mathscr{T}\) is nothing but the metric topology obtained from the discrete metric ...Can someone demonstrate/explain exactly how/why this is true ...?
Help will be much appreciated ...
Peter
===================================================================================Example 3.10 (b) above refers to Example 3.2 (a) ... ... so I am providing access to the same ... as follows:
View attachment 9212
It may help readers of the above post to have access to Stromberg's definition of a topological space ... so I am providing access to the same ... as follows:
View attachment 9209
Stromberg's definition of a topological space refers to Theorem 3.6 ... ... so I am providing access to the statement of the same ... as follows:
View attachment 9210
Stromberg's definition of a topological space also refers to Definition 3.3 ... ... so I am providing access to the same ... as follows:
View attachment 9211
Hope that helps ...
Peter
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand Example 3.10 (b) on page 95 ... ... Example 3.10 (b) reads as follows:
View attachment 9208
My question is as follows:Stromberg says that if \(\displaystyle X\) is any set and \(\displaystyle \mathscr{T}\) is the family of all subsets of \(\displaystyle X\) ...
... then \(\displaystyle \mathscr{T}\) is nothing but the metric topology obtained from the discrete metric ...Can someone demonstrate/explain exactly how/why this is true ...?
Help will be much appreciated ...
Peter
===================================================================================Example 3.10 (b) above refers to Example 3.2 (a) ... ... so I am providing access to the same ... as follows:
View attachment 9212
It may help readers of the above post to have access to Stromberg's definition of a topological space ... so I am providing access to the same ... as follows:
View attachment 9209
Stromberg's definition of a topological space refers to Theorem 3.6 ... ... so I am providing access to the statement of the same ... as follows:
View attachment 9210
Stromberg's definition of a topological space also refers to Definition 3.3 ... ... so I am providing access to the same ... as follows:
View attachment 9211
Hope that helps ...
Peter
Attachments

Stromberg  Example 3.10 (b) ... .png8 KB · Views: 93

Stromberg  Defn 3.9 ... Defn of a Topological Space ... .png19.3 KB · Views: 99

Stromberg  Statement of Theorem 3.6 ... .png5.2 KB · Views: 89

Stromberg  Defn 3.3 ... Defn of a Ball of Radius r with Center a ... .png7.9 KB · Views: 103

Stromberg  Example 3.2 (a) ... Discrete Metric Space .png14.6 KB · Views: 116