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

Peter

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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 a statement by Browder in Section 6.1 ... ...

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

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 ...

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 ... ...

Hope that helps ...

Peter

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HallsofIvy

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MHB Math Helper
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
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