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Stream: theory: mathematics

Topic: Metrizability for generalized metric spaces

Jean-Baptiste Vienney (Dec 16 2023 at 15:14):

Todd Trimble said:

Confirming that quasi-pseudo-metric spaces are the same thing as Lawvere metric spaces. "Quasi" is where you drop the symmetry condition $d(x, y) = d(y, x)$. "Pseudo" is where you drop the separation condition $d(x, y) = 0$ implies $x = y$.

Almost, you only have forgotten that for a Lawvere metric space, $d(x,y)\in [0,\infty]$. But choosing $[0,\infty]$ or $[0,\infty)$ is a bit arbitrary, we could as well choose any order commutative monoid $M$ and then a Lawvere $M$-metric space is just a category enriched over the poset $M$ with tensor product + and monoidal unit 0 (I think?).

Todd Trimble (Dec 16 2023 at 15:17):

You're right that I forgot. $[0, \infty]$ is a better choice (quantales have initial objects).

Jean-Baptiste Vienney (Dec 16 2023 at 15:19):

John Baez said:

Chris wrote:

It seems like one reason people may not have studied this is that for $T_1$ spaces, pseudometrizability agrees with metrizability.

I guess it's good to think about metrizability versus Lawvere metrizability for the space $\{x,y\}$ with just three open sets: $\{x\}$, $\{x,y\}$ and the empty set.

This topological space is interesting because it is not even not metrizable but also not pseudometrizable. But is it Lawvere metrizable? By the way, it’s difficult to choose what Lawvere metrizable means because there are two ways to define the open balls due to the absence of symmetry. I’ve read that people defined bitopological space s precisely for this reason: to be able to equip a quasi-metric space with these two topologies at the same time.

Jean-Baptiste Vienney (Dec 16 2023 at 15:21):

Chris Grossack (they/them) said:

This has definitely been studied, but it seems to be fairly niche. Here are some references I was able to find, though:

• Marín's *Weak Bases and Quasi-Pseudo-Metrization of Bispaces*
• Raghavan's notes
• This mse question about a "pseudometrization lemma"
• Minguzzi's *Quasi-Pseudo-Metrization of Topological Preordered Spaces*
• Li and Lin's *Characterizations of Quasi-Pseudo-Metrizable Spaces*.

It seems like one reason people may not have studied this is that for $T_1$ spaces, pseudometrizability agrees with metrizability (see here). Indeed another mse question here says that for $T_0$ spaces a pseudometric is a metric! (I think there may be multiple concepts called "pseudometric", so one may not be an immediate strengthening of the other. I haven't thought about this at all, I'm just sending you stuff I found, so I'm not sure precisely which definitions these answerers are using)

---

I hope this helps ^_^

Thanks a lot! I’ve been a bit lazy in my search thank your for your help ahah

John Baez (Dec 16 2023 at 16:11):

For any Lawvere metric $d$ on a set there's a Lawvere metric $d^\ast$ with $d^\ast(x,y) = d(y,x)$, and also a symmetric Lawevere metric $d+d^\ast$.

However, given a Lawvere metric space $X$, I'd be inclined to make it into a topological space where $U \subseteq X$ is open iff for each $x \in U$ and each $\epsilon > 0$ all points $y$ with $d(x,y) < \epsilon$ are in $U$.

I could copy this idea using $d^\ast$ or $d + d^\ast$, but it seems nicest to just use $d$.

John Baez (Dec 16 2023 at 16:12):

Then I believe the 2-point space I described is Lawvere metrizable.

Jean-Baptiste Vienney (Dec 16 2023 at 16:19):

Yes, I agree. This is very interesting as this is a common example of a non-pseudo-metrizable space. By the way, it is named the Sierpiński space.

John Baez (Dec 16 2023 at 16:22):

Yes, it's the "open set classifier": an open subset of any topological space $X$ is the same as a continuous map from $X$ to this 2-point space!

Jean-Baptiste Vienney (Dec 16 2023 at 16:23):

Ooh, ok!

Jean-Baptiste Vienney (Dec 16 2023 at 16:24):

Another example of non-pseudo-metrizable space is the cofinite topology of an infinite space $X$, where a set is closed iff it is finite or the whole space. I would be interested to know if it is Lawvere-metrizable ahah.

Jean-Baptiste Vienney (Dec 16 2023 at 16:27):

I'm a bit cheating because I'm currently studying for my point-set topology exam next week but the proof that this space is not pseudo-metrizable doesn't seem to work if we remove the symmetry requirement.

Jean-Baptiste Vienney (Dec 16 2023 at 16:31):

There has been work done on this!

Jean-Baptiste Vienney (Dec 16 2023 at 16:31):

Screenshot-2023-12-16-at-11.30.48AM.png