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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:
View attachment 9156
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 ...
View attachment 9158
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 ... ...
View attachment 9159
View attachment 9160
View attachment 9161Hope 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 a statement by Browder in Section 6.1 ... ...
The relevant statements by Browder follow Definition 6.10 and read as follows:
View attachment 9156
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 ...
View attachment 9158
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 ... ...
View attachment 9159
View attachment 9160
View attachment 9161Hope that helps ...
Peter
Attachments

Browder  1  Defn of Interior 6.10 and Relevant Remarks ... PART 1 ... ....png11.2 KB · Views: 97

Browder  2  Defn of Interior 6.10 and Relevant Remarks ... PART 2 ... .png3.7 KB · Views: 86

Browder  Defn of Closure 6.7 and Relevant Propn 6.8 ... .png21 KB · Views: 86

Browder  1  Start of 6.1  Relevant Defns & Propns ... PART 1 ... .png15.4 KB · Views: 68

Browder  2  Start of 6.1  Relevant Defns & Propns ... PART 2 ... .png18.7 KB · Views: 90

Browder  3  Start of 6.1  Relevant Defns & Propns ... PART 3 ... .png56.3 KB · Views: 96
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