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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:

" ... ... \(\displaystyle x \in \overline{E}\) if and only if \(\displaystyle U \cap E \neq \emptyset\) for every open neighborhood \(\displaystyle U\) of \(\displaystyle x\) (and hence for every neighborhood \(\displaystyle U\) of \(\displaystyle x\)) ... ..."My question is as follows:

Why, if the statement: " ... \(\displaystyle x \in \overline{E}\) if and only if \(\displaystyle U \cap E \neq \emptyset\) .. "

... is true for every open neighborhood \(\displaystyle U\) of \(\displaystyle x\) ...

... is the statement necessarily true for every neighborhood \(\displaystyle U\) of \(\displaystyle x\) ... ?

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:

" ... ... \(\displaystyle x \in \overline{E}\) if and only if \(\displaystyle U \cap E \neq \emptyset\) for every open neighborhood \(\displaystyle U\) of \(\displaystyle x\) (and hence for every neighborhood \(\displaystyle U\) of \(\displaystyle x\)) ... ..."My question is as follows:

Why, if the statement: " ... \(\displaystyle x \in \overline{E}\) if and only if \(\displaystyle U \cap E \neq \emptyset\) .. "

... is true for every open neighborhood \(\displaystyle U\) of \(\displaystyle x\) ...

... is the statement necessarily true for every neighborhood \(\displaystyle U\) of \(\displaystyle x\) ... ?

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