Space Suggestion: Use "Fréchet space R^ω" as the default name for S30

Space Suggestion

Use "Fréchet space $\mathbb R^\omega$" as the default name for S30.

Rationale

While there are numerous aliases and characterizations for this space, the most elegant (in terms of the topological properties we model in $\pi$-Base) is probably the Tychonoff product of countably-many copies of the real line.

See also discussion at #956.

It would be a little bit of a shame to lose the Hilbert space from the main list, but your point has some validity.
Before making a decision, note also that most of the proofs for the asserted traits assume the Hilbert space incarnation.

Soliciting more input from other contributors.

There is technically some confusion about the name "Fréchet space", coming from someone with a background in functional analysis. That confusion is sort of already there in the description for S30, namely, in the line "... any infinite-dimensional separable Banach space or separable Fréchet space is homeomorphic to...", where the term "Fréchet space" means very different things than $\mathbb{R}^\omega$. Instead, it refers to a locally convex topological vector space whose topology is completely metrizable. But I do agree that, from a purely topological perspective, $\mathbb{R}^\omega$ is sort of the most elegant one. Whether causing confusion for functional analysts is an important concern or not is debatable.

Have just $\mathbb R^\omega$ in the title?

I would also prefer to not have "Frechet space" in the title if we go with that space for the name.

@prabau

@prabau I don’t think any space in pi-Base currently has just symbols as its name. There’s always some English description attached. Maybe we can say “Countable product of reals $\mathbb{R}^\omega$”, parallel with S107?

@david20000813

I like @david20000813's last suggestion, "Countable product of reals $\mathbb{R}^\omega$".

I guess this was assumed, but 'Hilbert space $\ell^2$' should still at least be an alias if it's not the title.

And if we go with the suggested space as a main name, I guess for the proofs of traits that rely on Hilbert space, we can either replace them with an easy direct proof for $\mathbb{R}^\omega$ if there is one, or add "In its incarnation as the separable Hilbert space $\ell^2$, blah blah" or something of the kind, relying on the homeomorphisms presented in the README file.

"Countable product of reals $\mathbb R^\omega$" sounds great to me. Note Frechet also means $T_1$.

I'll make a PR for everyone's consideration.

Er, well, we should wait for #975 first (there's no rush here).

#975 is now complete.

Note Frechet also means $T_1$.

My training as a functional analyst instead of a topologist does show here lol. I only thought of Fréchet space as a kind of topological vector space instead of it being a synonym of $T_1$.

After further research, I realized there is even one more point of confusion: Fréchet Urysohn spaces were called Fréchet spaces in some sources.

After further research, I realized there is even one more point of confusion: Fréchet Urysohn spaces were called Fréchet spaces in some sources.

Ooh, we should add Fréchet as an alias here; I've see this as well.

@StevenClontz

@StevenClontz This paper mentions three different names for Fréchet Urysohn spaces: Fréchet spaces, closure-sequential spaces, and neighborhood-sequential (or $N$-sequential) spaces.

(And the Wikipedia article for sequential spaces, for some reason, mentions the names $N$-sequential and Fréchet Urysohn in the same section, without ever saying they are the same thing.)

After further research, I realized there is even one more point of confusion: Fréchet Urysohn spaces were called Fréchet spaces in some sources.

Ooh, we should add Fréchet as an alias here; I've see this as well.

How about not adding Frechet as an alias for Frechet-Urysohn? Someone who types Frechet in a search will see Frechet-Urysohn displayed already, so the clear choice to explore will be obvious.

But maybe we should add a hyphen: Fréchet-Urysohn? Seems to be more common.

I like the switch to Fréchet-Urysohn for the main name.

I still think adding the alias Fréchet is wise: if someone's reading an older paper and sees "Fréchet" they may not think to look at Fréchet-Urysohn without the alias.

I'll mock up a PR and we can see how we feel.

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Space Suggestion: Use "Fréchet space R^ω" as the default name for S30 #986