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 , 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 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)