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- Jun 22, 2012

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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

======================================================================================

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

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

======================================================================================

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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