Understanding Topology: Closure, Boundary & Open/Closed Sets

[Math Amateur]

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:

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View attachment 9157In 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======================================================================================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 ...

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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 ... ...

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View attachment 9161Hope that helps ...

Peter

 

[HallsofIvy]

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.
​

 

[Math Amateur]

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