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I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ...
I am focused on Chapter 2: The Rudiments of Plane Topology ...
I need help with some aspects of the proof of Lemma 3.1 ...
Lemma 3.1 and its proof read as follows:View attachment 7373
View attachment 7374
In the above text from Palka Section 2.2 we read the following:" ... ... The sets $$S$$ and $$T$$ are non-empty - $$S$$ contains $$A \cap U^*$$ and $$T$$ contains $$A \cap V^* $$ - and disjoint. ... ... "I am trying to show/demonstrate rigorously that the sets $$S$$ and $$T$$ are non-empty - $$S$$ contains $$A \cap U^*$$ and $$T$$ contains $$A \cap V^*$$ - and disjoint ... can someone please help ...?
Help will be much appreciated ...
Peter===================================================================================Readers of the above post will be assisted by having access to Palka's introduction to disconnected sets which includes the key definition ... so I am providing the same ... as follows ... :https://www.physicsforums.com/attachments/7375
I am focused on Chapter 2: The Rudiments of Plane Topology ...
I need help with some aspects of the proof of Lemma 3.1 ...
Lemma 3.1 and its proof read as follows:View attachment 7373
View attachment 7374
In the above text from Palka Section 2.2 we read the following:" ... ... The sets $$S$$ and $$T$$ are non-empty - $$S$$ contains $$A \cap U^*$$ and $$T$$ contains $$A \cap V^* $$ - and disjoint. ... ... "I am trying to show/demonstrate rigorously that the sets $$S$$ and $$T$$ are non-empty - $$S$$ contains $$A \cap U^*$$ and $$T$$ contains $$A \cap V^*$$ - and disjoint ... can someone please help ...?
Help will be much appreciated ...
Peter===================================================================================Readers of the above post will be assisted by having access to Palka's introduction to disconnected sets which includes the key definition ... so I am providing the same ... as follows ... :https://www.physicsforums.com/attachments/7375