# Neighbourhoods and Open Neighbourhoods .... Browder, Proposition 6.8 .... ....

• MHB
• Math Amateur
In summary, Peter was reading Andrew Browder's book, "Mathematical Analysis: An Introduction" and was having difficulty understanding an aspect of Proposition 6.8. He asked for help and received it from HallsofIvy and Opalg.
Math Amateur
Gold Member
MHB
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

#### Attachments

• Browder - Defn of Closure 6.7 and Relevant Propn 6.8 ... .png
21 KB · Views: 85
• Browder - 2 - Start of 6.1 - Relevant Defns & Propns ... PART 2 ... .png
18.7 KB · Views: 69
Your question is "why does $x\in \overline E$ and U an open imply $U\cap E$ non-empty"

Okay, $\overline{E}$ is the "closure of E", the intersection of all closed sets that contain E. And a set is "closed" if it contains all of its boundary points with a "boundary point" being a point, p, such that every open set containing p also contains at least one point of E.

Peter said:
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 $$\displaystyle x$$ ... ?
If $W$ is a neighbourhood of $x$ then (by definition) it contains an open neighbourhood of $x$. That open neighbourhood has a nonempty intersection with $E$, hence so does $W$.

Opalg said:
If $W$ is a neighbourhood of $x$ then (by definition) it contains an open neighbourhood of $x$. That open neighbourhood has a nonempty intersection with $E$, hence so does $W$.
HallsofIvy and Opalg ... thanks for the help

Peter

## 1. What is the definition of a neighbourhood in mathematics?

A neighbourhood in mathematics is a subset of a topological space that contains an open set containing a given point. It is often represented as a small open ball around the point.

## 2. How is a neighbourhood different from an open neighbourhood?

A neighbourhood is a general term for a set containing an open set around a point, while an open neighbourhood specifically refers to an open set containing the point.

## 3. What is the significance of open neighbourhoods in topology?

Open neighbourhoods are important in topology because they allow for the definition of continuity and convergence in topological spaces. They also help in defining the concept of a limit point.

## 4. What is Browder's Proposition 6.8 in neighbourhoods and open neighbourhoods?

Browder's Proposition 6.8 states that if a topological space has a countable basis, then every open neighbourhood of a point contains a neighbourhood of that point.

## 5. How does Proposition 6.8 impact the study of neighbourhoods in topology?

Proposition 6.8 is important in topology as it allows for the simplification of proofs and theorems by reducing the number of open sets that need to be considered. It also helps in understanding the relationship between open sets and neighbourhoods in topological spaces.

Replies
2
Views
2K
Replies
5
Views
2K
Replies
2
Views
1K
Replies
5
Views
860
Replies
2
Views
2K
Replies
2
Views
1K
Replies
2
Views
2K
Replies
2
Views
1K
Replies
3
Views
2K
Replies
2
Views
1K