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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am reading Chapter 6: Topology ... ... and am currently focused on Section 6.1 Topological Spaces ...
I need some help in order to fully understand an aspect of Proposition 6.8 ... ...
Proposition 6.8 (and the relevant Definition 6.8 ... ) read as follows:
View attachment 9163In the above text (in the statement of Proposition 6.8 ...) we read the following:
" ... ... $$x \in \overline{E}$$ if and only if $$U \cap E \neq \emptyset$$ for every open neighborhood $$U$$ of $$x$$ (and hence for every neighborhood $$U$$ of $$x$$) ... ..."My question is as follows:
Why, if the statement: " ... $$x \in \overline{E}$$ if and only if $$U \cap E \neq \emptyset$$ .. "
... is true for every open neighborhood $$U$$ of $$x$$ ...
... is the statement necessarily true for every neighborhood $$U$$ of $$x$$ ... ?
Help will be appreciated ...
Peter=====================================================================================The definition of a neighborhood is relevant to the above post ... so I am providing access to Browder's definition of the same as follows:
View attachment 9164
Hope that helps ...
Peter
I am reading Chapter 6: Topology ... ... and am currently focused on Section 6.1 Topological Spaces ...
I need some help in order to fully understand an aspect of Proposition 6.8 ... ...
Proposition 6.8 (and the relevant Definition 6.8 ... ) read as follows:
View attachment 9163In the above text (in the statement of Proposition 6.8 ...) we read the following:
" ... ... $$x \in \overline{E}$$ if and only if $$U \cap E \neq \emptyset$$ for every open neighborhood $$U$$ of $$x$$ (and hence for every neighborhood $$U$$ of $$x$$) ... ..."My question is as follows:
Why, if the statement: " ... $$x \in \overline{E}$$ if and only if $$U \cap E \neq \emptyset$$ .. "
... is true for every open neighborhood $$U$$ of $$x$$ ...
... is the statement necessarily true for every neighborhood $$U$$ of $$x$$ ... ?
Help will be appreciated ...
Peter=====================================================================================The definition of a neighborhood is relevant to the above post ... so I am providing access to Browder's definition of the same as follows:
View attachment 9164
Hope that helps ...
Peter