Talk:Continuous Surjection Induces Continuous Bijection from Quotient Space/Corollary 1

I think there's a good argument for:

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces

that is, keeping the $T_1$ and $T_2$.

Reason: you would then discuss:

Let $g: T_1 \to T_2$ be a continuous surjection.

Continuity is not relevant until you take the topology into account. Hence you need to specify the mapping as to be from and to the structures that include that topology.

Otherwise if $S_1 = S_2$ (and they only differ by their topologies) then saying: "Let $g: S_1 \to S_1$ ... etc. loses its meaning.

Thoughts? --prime mover (talk) 22:39, 23 September 2022 (UTC)

The (probably) non-standard notation $g: T_1 \to T_2$ is not yet defined in Definition:Continuous Mapping (Topology). In my opinion, even if $S_1 = S_2$, writing $g: S_1 \to S_2$ clearly indicates the topologies to consider.--Usagiop (talk) 23:48, 23 September 2022 (UTC)

How about what I've done? I believe, since there are possible conceptual subtleties here, that it may be important to be specific as to exactly what sort of continuity is being considered here (in particular, Definition:Continuous Mapping is such a wide concept that it's better to link to the specific instance that is appropriate. --prime mover (talk) 08:08, 24 September 2022 (UTC)

How about the continuity of $f$?--Usagiop (talk) 13:39, 24 September 2022 (UTC)

See the parent page. --prime mover (talk) 19:46, 24 September 2022 (UTC)

The simple reason why I removed the symbols $T_1$ and $T_2$ is because they were not used in the proof of the theorem; you can put them back if you want.

The usual convention is if there is only one topology $\tau_1$ defined on $S_1$ and only one topology $\tau_2$ defined on $S_2$, the statement '$g: S_1 \to S_2$ is continuous' silently assumes that we are talking about $\struct {\tau_1 , \tau_2 }$-continuity. This assumption is commonly used in $\mathsf{Pr} \infty \mathsf{fWiki}$, as far as I can see; for instance, in Definition:Path (Topology), we read:

A path in $T$ is a continuous mapping $\gamma: I \to S$.

when the actual meaning is something like:

Let $\tau_I$ be the subspace topology on $I$ induced by the Euclidean topology on $\R$.

A path in $T$ is a $\struct{ \tau_I ,\tau}$-continuous mapping $\gamma: I \to S$.

To me, the later version sacrifices the brevity and does not significantly improve the clarity of the statement.

When you're considering 3 spaces, one of them being the quotient of one of the other ones, it does significantly improve the clarity, particularly when you are new to the subject and are unfamiliar with the nature of the objects being dealt with. But of course we don't care about that, we're only writing for the benefit of those who are already experts. (Irony.) --prime mover (talk) 19:45, 24 September 2022 (UTC)

But seriously, how much does it improve the clarity for learners unfamiliar with the subject if we make them read expressions like '$\struct { \tau_X, \tau_Y } $-continuous' that are not widely used elsewhere? I do realize that this particular expression is already established on $\mathsf{Pr} \infty \mathsf{fWiki}$, and I certainly do not want to change our conventions. But I cannot find any other webpage or text book that needs to explain continuity like that. --Anghel (talk) 23:11, 24 September 2022 (UTC)

My bad. Fixed. --prime mover (talk) 06:56, 25 September 2022 (UTC)

I think even the quotient has a unique topology. There seems no confusion. I am lost.--Usagiop (talk) 23:30, 24 September 2022 (UTC)

The whole point of this site is to educate. Some people have difficulty with this stuff. I am one of those people. IT HELPS. --prime mover (talk) 06:56, 25 September 2022 (UTC)

I just wanted to understand. The first example with $S_1=S_2$ was indeed confusing. But I cannot get the point of the second example with 3 spaces including a quotient. Writing down an explicit example could help me. --Usagiop (talk) 09:29, 25 September 2022 (UTC)

But it is a non-trivial question how to write about the topologies in use. In the latest theorem I put up, Simple Loop in Hausdorff Space is Homeomorphic to Quotient Space of Interval, I did not specify the topology on $\closedint 0 1$, but I defined all other topologies in use.

As for what subpage of Definition:Continuous Mapping to link to, I must admit I usually link to the main page, because all definitions of continuity implies the Definition:Continuous Mapping (Topology). I will try to make more specific links. --Anghel (talk) 15:10, 24 September 2022 (UTC)

It Was Decided some time back (2017-18 or so, not by me) that the most precise definition page possible was used in all cases. I have since then tried to enforce that policy as ordered. Might rethink as it's tiresome. --prime mover (talk) 19:42, 24 September 2022 (UTC)

It is mainly you who enforces the standards, so do not worry about changing them. --Anghel (talk) 23:11, 24 September 2022 (UTC)

That's because nobody else will. Right, I'm done. To the bin with the standards. Everybody can post up whatever they want in whatever format they want. It doesn't matter. Nothing matters. We all die in the end. --prime mover (talk) 06:31, 25 September 2022 (UTC)

You must be missing an irony tag here, as you must derive some kind of joy from looking over us editors. Otherwise, you would not log on every day, unless you consider life to be so truly horrible, that there is nothing better to do than edit the $\mathsf{Pr} \infty \mathsf{fWiki}$. --Anghel (talk) 23:00, 25 September 2022 (UTC)

There is no joy in life, merely duty, responsibility and pain. --prime mover (talk) 05:38, 26 September 2022 (UTC)

Well, you have neither any duty nor responsibility for the $\mathsf{Pr} \infty \mathsf{fWiki}$ (other than the sunk cost of having edited the site for +10 years), so I assume you must get something out of it, even if it is just having a place to complain about life. --Anghel (talk) 13:41, 27 September 2022 (UTC)