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Boundary of a Set in a Topological Space ... Browder, Remarks Following Defn 6.10, pages 125-126 ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
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 a statement by Browder in Section 6.1 ... ...


The relevant statements by Browder follow Definition 6.10 and read as follows:




Browder - 1 - Defn of Interior 6.10 and Relevant Remarks ... PART 1 ... ....png
Browder - 2 - Defn of Interior 6.10 and Relevant Remarks ... PART 2 ... .png





In the above text we read the following:

" ... ... The set \(\displaystyle \overline{E}\) \ \(\displaystyle E^{ \circ }\) is referred to as the boundary of \(\displaystyle E\), and is denoted by \(\displaystyle \text{bdry } E\); it is easy to see that \(\displaystyle \text{bdry } E = \emptyset\) if and only if \(\displaystyle E\) is both open and closed ... ... "


My question is as follows:

Can someone explain and demonstrate rigorously how/why \(\displaystyle \text{bdry } E = \emptyset\) if and only if \(\displaystyle E\) is both open and closed ... ... ?



Note: I was surprised at Browder's remark above because I was always under the impression (delusion?) that an open set had an empty set as its boundary ...



Help will be appreciated ... ...

Peter


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The closure of a subset of a topological space is an important notion in the above post so I am providing Browder's definition of closure as well as a basic proposition involving closure ... as follows ...




Browder - Defn of Closure 6.7 and Relevant Propn 6.8  ... .png




It may help readers of the above post to have access to Browder's fundamental topological definitions ... so I am providing the same as follows ... ...




Browder - 1 - Start of 6.1 - Relevant Defns & Propns ... PART 1 ... .png
Browder - 2 - Start of 6.1 - Relevant Defns & Propns ... PART 2 ... .png
Browder - 3 - Start of 6.1 - Relevant Defns & Propns ... PART 3 ... .png





Hope that helps ...

Peter
 
Last edited:

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
Note: I was surprised at Browder's remark above because I was always under the impression (delusion?) that an open set had an empty set as its boundary …


I can't say whether you are deluded or not but you are certainly wrong! Consider the "open interval" (0, 1) in R. Its closure is the closed set [0, 1]. Its boundary is {0, 1}.

What is true of an open set is that "an open set contains none of its boundary points", not that has no boundary points.

A closed set, conversely contains all of its boundary points. A set is both "open" and "closed" if and only if "none" is the same as "all". That is, if the set has no boundary points.
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
I can't say whether you are deluded or not but you are certainly wrong! Consider the "open interval" (0, 1) in R. Its closure is the closed set [0, 1]. Its boundary is {0, 1}.

What is true of an open set is that "an open set contains none of its boundary points", not that has no boundary points.

A closed set, conversely contains all of its boundary points. A set is both "open" and "closed" if and only if "none" is the same as "all". That is, if the set has no boundary points.
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Oh yes ... of course ... you're right ...

Thanks for the clarification... it was most helpful...

Peter