 Thread starter
 #1
I am reading Stephen Willard: General Topology ... ... and am studying Chapter 2: Topological Spaces and am currently focused on Section 3: Fundamental Concepts ... ...
I need help in order to prove the assertion in Willard Exercise 3B ...
The assertion in Willard Exercise 3B reads as follows:
I am assuming that Willard is assuming the usual topology and metric in \(\displaystyle \mathbb{R}^2\) ... and so a set \(\displaystyle E\) is open iff for each \(\displaystyle x \in E\) there is an \(\displaystyle \epsilon\)disk (open ball) \(\displaystyle U(x, \epsilon)\) about x contained in \(\displaystyle E\) ...
Can someone please rigorously demonstrate the truth of assertion 3B ...
Help will be much appreciated ...
Peter
===============================================================================================================
Readers of the above post may be helped by access to Willard's definition of an \(\displaystyle \epsilon\)disk (open ball) and an open set in a metric space so I am providing access to the same ... as follows:
Readers of the above post may also be helped by access to Willard's definition of frontier (boundary) and some basic results regarding the frontier of a set .... so I am providing access to the same ... as follows: ...
Readers of the above post may also be helped by access to Willard's definition of closure and some basic results regarding the closure of a set .... so I am providing access to the same ... as follows: ...
Readers of the above post may also be helped by access to Willard's definition of interior and some basic results regarding theinterior of a set .... so I am providing access to the same ... as follows: ...
Hope that helps ...
Peter
I need help in order to prove the assertion in Willard Exercise 3B ...
The assertion in Willard Exercise 3B reads as follows:
I am assuming that Willard is assuming the usual topology and metric in \(\displaystyle \mathbb{R}^2\) ... and so a set \(\displaystyle E\) is open iff for each \(\displaystyle x \in E\) there is an \(\displaystyle \epsilon\)disk (open ball) \(\displaystyle U(x, \epsilon)\) about x contained in \(\displaystyle E\) ...
Can someone please rigorously demonstrate the truth of assertion 3B ...
Help will be much appreciated ...
Peter
===============================================================================================================
Readers of the above post may be helped by access to Willard's definition of an \(\displaystyle \epsilon\)disk (open ball) and an open set in a metric space so I am providing access to the same ... as follows:
Readers of the above post may also be helped by access to Willard's definition of frontier (boundary) and some basic results regarding the frontier of a set .... so I am providing access to the same ... as follows: ...
Readers of the above post may also be helped by access to Willard's definition of closure and some basic results regarding the closure of a set .... so I am providing access to the same ... as follows: ...
Readers of the above post may also be helped by access to Willard's definition of interior and some basic results regarding theinterior of a set .... so I am providing access to the same ... as follows: ...
Hope that helps ...
Peter
Attachments

60 KB Views: 1
Last edited: