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Stream: theory: mathematics

Topic: Metrizability for genetralized metric spaces

Jean-Baptiste Vienney (Dec 16 2023 at 02:22):

Metrizability is (or was) a very popular topic in point-set topology. When I google it, I can't find anything about metrizability where the notion of metric is weakened. For instance pseudometrizability where a pseudometric is like a metric but without $\rho(x,y)=0 \Rightarrow x=y$. Or Lawvere-metrizability. Does anyone know something or references about that?

Chris Grossack (they/them) (Dec 16 2023 at 06:50):

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 ^_^

Chris Grossack (they/them) (Dec 16 2023 at 06:58):

Ah! I clearly should have looked up "lawvere metrization" as well as "pseudometrization". When I did that I immediately found this mo question which claims that every topological space with a countable base is lawvere metrizable (it also links to another mse post which might itself be worth a read!)

This might be implicit in some of the posts I linked above (again, I didn't really read them) but it's nice to see it spelled out.

Todd Trimble (Dec 16 2023 at 11:00):

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 $x = y$.

John Baez (Dec 16 2023 at 11:37):

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.

Todd Trimble (Dec 16 2023 at 11:42):

Indeed another mse question here says that for $T_0$ spaces a pseudometric is a metric!

Of course what Brian Scott in his answer meant is that a pseudometrizable $T_0$ space is metrizable, using the same (pseudo)metric. The proof is not difficult.

Todd Trimble (Dec 16 2023 at 11:42):

Pseudometric spaces typically crop up in analysis when people are studying the topology induced by a seminorm. Seminorms crop up a lot in functional analysis. For example, if $U$ is an open region in $\mathbb{R}^n$, then one can write $U$ as a countable union of compact subsets $K_n$, and consider for continuous functions $f: U \to \mathbb{C}$ the function

$\rho_n(f) = \max_{x \in K_n} |f(x)|$

which is a seminorm but not yet a norm (but the $\rho_n$ collectively form a separating family of seminorms, and there is a standard way of creating a norm from such a separating family). Variations on this theme are used to create suitable topologies on spaces of smooth functions on open regions.