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Martin Sleziak
Martin Sleziak
1:28 AM
An example of $m$ please — youssef sabar 1 min ago
What about this one?
$$
\begin{array}{c|ccc}
& a & b & c \\\hline
a & 0 & 0 & 1 \\
b & 0 & 1 & 1 \\
c & 1 & 1 & 2
\end{array}
$$
You should be able to find at least one example of $m$ based on the finite example in the linked question. Maybe somebody else will have some other suggestions - if you wish, we can also discuss this sometimes either in general topology chatroom or in my chatroom. — Martin Sleziak 9 secs ago
I will add that it's almost 3 a.m. in my timezone, so i am going to get some sleep. But I will notice if you leave here - or in the other room - some (constructive) messages and if I have time, I will try to get back to them.
Maybe it will be useful to remind that instructions how to use MathJax in SE chat can be found here and here.
The above is related to this question:
-3
Q: Base of topology for metric-like space

youssef sabarLet $X$ be a nonempty set and $p:X\times X\rightarrow\mathbb{R}^+ $ be a function satisfying the following conditions for all $x,y,z\in X$: \begin{align} &1)\enspace p(x,y)=0\implies x=y \\ &2)\enspace p(x,y)=p(y,x)\hspace{1,2cm}\\ \hspace{0,2cm}&3)\enspace p(x,z)\leq p(x,y)+p(y,z) \end{align} Th...

gn.general-topology metric-spaces fixed-point-theorems
 
 
4 hours later…
Martin Sleziak
Martin Sleziak
5:22 AM
@youssefsabar In case it helps, I have also asked this question on math.SE: When does a symmetric function give a base for a topology?
Maybe somebody posts there something interesting which might help you to get to some further counterexamples.
@youssefsabar Only now I realized that you cannot talk in chat since you are below 20 reputation points. (To be more precise, you cannot talk in chat without help of moderators - at least until you gain sufficient reputation on one of the sites.) Still you can read transcripts, maybe some of the comments I've made in chat might be useful for you. — Martin Sleziak 19 secs ago
 
 
2 hours later…
Martin Sleziak
Martin Sleziak
7:51 AM
@MartinSleziak Correction, the example was supposed to be this:
$$
\begin{array}{c|ccc}
& a & b & c \\\hline
a & 0 & 0 & 1 \\
b & 0 & 0 & 0 \\
c & 1 & 0 & 0
\end{array}
$$
If we define $m$ in this way, both $B(a,1/2)=\{a,b\}$ and $B(c,1/2)=\{c,b\}$ are balls but their intersection is not open.
 

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