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The Trivial or Indiscrete Topology is Pseudometrizable ... Willard, Example 3.2(d) ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,886
Hobart, Tasmania
I am reading Stephen Willard: General Topology ... ... and am currently focused on Chapter 1: Set Theory and Metric Spaces and am currently focused on Section 2: Metric Spaces ... ...

I need help in order to fully understand Example 3.2(d) ... ..


Example 3.2(d) reads as follows:

Willard - Example 3.2 (d) ... .png

and Example 2.7(e) reads as follows:

Willard - Example 2.7 (e) ... .png


In Example 3.2(d) we read the following:

" ... It is pseudometrizable since it is the topology generated by the trivial pseudometric on X, by part (e) of Example 2.7. ... ... "


I am somewhat lost by this example ...

Can someone please demonstrate that (X, \(\displaystyle \tau\) ) is the topology generated by the trivial pseudometric on X ... and explain the relation to part (e) of Example 2.7. ...


Help will be much appreciated ...

Peter
 

GJA

Well-known member
MHB Math Scholar
Jan 16, 2013
255
Hi Peter ,

The statement, "It is pseudometrizable since it is the topology generated by the trivial pseudometric on $X$" (i.e., without the reference to Example 2.7(e)) is true because the indiscrete topology on $X$ is the same as the topology on $X$ generated by the pseudometric defined by $d(x,y) = 0$ for all $x,y\in X$. Considering the fact that one point sets in $X$ are not closed when $X$ is equipped with the indiscrete topology (and $X$ contains more than one point), the reference to 2.7(e) is very likely a typo.
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,886
Hobart, Tasmania
Hi Peter ,

The statement, "It is pseudometrizable since it is the topology generated by the trivial pseudometric on $X$" (i.e., without the reference to Example 2.7(e)) is true because the indiscrete topology on $X$ is the same as the topology on $X$ generated by the pseudometric defined by $d(x,y) = 0$ for all $x,y\in X$. Considering the fact that one point sets in $X$ are not closed when $X$ is equipped with the indiscrete topology (and $X$ contains more than one point), the reference to 2.7(e) is very likely a typo.
Thanks GJA ...

Your post clarifies the issue ... previously I was very confused by the reference to 2.7(e) ...

Thanks again,

Peter