1
$\begingroup$

Let $X$ 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} Then the pair $(X,p)$ is said to be a metric-like space.

I want to show please that each metric-like $p$ on $X$ generates a topology $τ_p$ on $X$ whose base is the family of open-balls $$B(x,\varepsilon)=\{y\in X:|p(x,y)-p(x,x)|<\varepsilon\}.$$

Thank you.

$\endgroup$
10
  • $\begingroup$ There is a typo in the question that makes it difficult to understand what is being asked, but if one replaces the $d$'s with $p$'s in the bottom part of the question, then an interesting question emerges. (@youssef: I think, but I'll have to check details, that the answer is no in general, but yes if $p$ is continuous.) I'm voting to reopen. $\endgroup$
    – Will Brian
    Dec 6, 2017 at 17:50
  • 1
    $\begingroup$ @WillBrian As a follow-up to your edit I have also corrected a few minor typos and explicitly added to the post that this is different from metric space. (Since this can be missed if somebody does not read carefully.) A quick Google search leads to the paper A. Amini-Harand: Metric-like spaces, partial metric spaces and fixed points, doi.org/10.1186/1687-1812-2012-204. $\endgroup$ Dec 6, 2017 at 18:02
  • $\begingroup$ @ Arturo Magidin, does not reearch-level questions?!! $\endgroup$ Dec 6, 2017 at 19:54
  • 1
    $\begingroup$ It seems that Wikipedia calls this a metametric. A reference given there is Väisälä, Jussi (2005), "Gromov hyperbolic spaces", Expositiones Mathematicae, 23 (3): 187–231, doi: 10.1016/j.exmath.2005.01.010. From this paper: "A metametric space is metrizable. In fact, a metametric $d$ can be changed to a metric $d_1$ simply by setting $d(x,x)=0$ and $d_1(x,y)=d{x,y}$ for $x\ne y$. Then $d$ and $d_1$ define the same topology." $\endgroup$ Dec 7, 2017 at 1:00
  • 1
    $\begingroup$ @MartinSleziak I have cast the final vote to reopen, so that you can put some of the details from your earlier comments into an answer below $\endgroup$
    – Yemon Choi
    Dec 11, 2017 at 1:04

1 Answer 1

3
$\begingroup$

Based on Yemon Choi's suggestion I am posting here an answer. So far it is mostly a summary of stuff which is was said in a post on another site and in the comments above. But if you have something to add (and it is not enough for a separate answer), feel free to edit this. (After all, this is community wiki.)

A quick Google search leads to the paper A. Amini-Harand: Metric-like spaces, partial metric spaces and fixed points, doi.org/10.1186/1687-1812-2012-204. This paper contains a definition of metric-like space in the same way as given in the question and contains a claim that the open balls, defined as above, indeed give a topology (without a proof).

Counterexamples

Related notions

  • As mlk points out in their answer, there is a related notion of patrial metric, which is also mentioned in Amini-Harand's paper. One of the reason for the problems might be that a different version of triangle inequality is needed. The definition of partial metric requires:
    • $x=y$ iff $p(x,y)=p(x,x)=p(x,y)$
    • $0\le p(x,x) \le p(x,y)$
    • $p(x,y)=p(y,x)$
    • $p(x,z) \le p(x,y)+p(y,z)-p(x,x)$
  • The Wikipedia article on metric (current revision) contains a definition of metametric which is exactly the same as the above definition of metric-like function. The reference given there is: Väisälä, Jussi (2005), "Gromov hyperbolic spaces", Expositiones Mathematicae, 23 (3): 187–231, doi: 10.1016/j.exmath.2005.01.010. However, the topology in this paper is defined differently. (For example, a point $x$ is isolated whenever $p(x,x)>0$.)
  • The Wikipedia article on metric also defines the notion of premetric (current revision) where only conditions $d(x,x)=0$ and $d(x,y)\ge0$ are required. (Including the warning that this is not a standard term and terminology can vary). Clearly, $d(x,y)=|p(x,y)-p(x,x)|$ is a premetric. According to the Wikipedia article, every premetric gives a topology but in this way: A set $U$ is open if for every $x\in U$ there exist some ball $B(x,\varepsilon)=\{y\in X; d(x,y)<\varepsilon\}$ with $B(x,\varepsilon)\subseteq U$. It is explicitly mentioned that these balls are not necessarily open. (So the balls described here are not necessarily a base. And the topology is obtained from these balls in a different way than described in the linked paper.) This way of obtaining a topology is analogous to the way a topology is obtained from a metametric in Väisälä's paper.
$\endgroup$
3
  • $\begingroup$ An example of $m$ please $\endgroup$ Dec 12, 2017 at 1:27
  • $\begingroup$ 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. $\endgroup$ Dec 12, 2017 at 1:35
  • $\begingroup$ @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. $\endgroup$ Dec 12, 2017 at 5:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.