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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 the proof of Theorem 3.36 on page 102 ... ... Theorem 3.36 and its proof read as follows:
View attachment 9134
In the above proof by Stromberg we read the following:
" ... ...Next let $$U = \bigcap_{ k = 1 }^n U_{ y_k }$$. Then $$U$$ is a neighbourhood of $$x$$ and $$U \subset S'$$ ... "
My question is as follows:
It seems plausible that $$U \subset S'$$ ... ...
... ... BUT ... ...
... how would we demonstrate rigorously that $$U \subset S'$$ ... ... ?
(Note that $$S'$$ is $$S$$ complement ...)
Help will be much appreciated ... ...
Peter=================================================================================The above post mentions Hausdorff spaces ... so I am providing access to Stromberg's definition of a Hausdorff space ... as follows:
View attachment 9135I believe it may be helpful to MHB readers to have access to some of Stromberg's terminology and notation associated with topological spaces ... so I am providig access to the same ... as follows:
View attachment 9136
Hope that helps ... ...
Peter
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Theorem 3.36 on page 102 ... ... Theorem 3.36 and its proof read as follows:
View attachment 9134
In the above proof by Stromberg we read the following:
" ... ...Next let $$U = \bigcap_{ k = 1 }^n U_{ y_k }$$. Then $$U$$ is a neighbourhood of $$x$$ and $$U \subset S'$$ ... "
My question is as follows:
It seems plausible that $$U \subset S'$$ ... ...
... ... BUT ... ...
... how would we demonstrate rigorously that $$U \subset S'$$ ... ... ?
(Note that $$S'$$ is $$S$$ complement ...)
Help will be much appreciated ... ...
Peter=================================================================================The above post mentions Hausdorff spaces ... so I am providing access to Stromberg's definition of a Hausdorff space ... as follows:
View attachment 9135I believe it may be helpful to MHB readers to have access to some of Stromberg's terminology and notation associated with topological spaces ... so I am providig access to the same ... as follows:
View attachment 9136
Hope that helps ... ...
Peter