How to characterize the net convergence in final topology?

Question

Given a set $X$ and an indexed family $\{(Y_i,\mathscr T_i)\}_{i\in I}$ of topological spaces with functions $f_i:Y_i\to X$. Let $\tau_{\text{final}}$ be the final topology in $X$ w.r.t. $\{f_i\}_{i\in I}$, that is, it's the finest topology such that each $f_i:(Y_i,\mathscr T_i)\to (X,\tau_{\text{final}})$ is continuous.

Now consider a net $\{x_\alpha\}_{\alpha\in A}\subset X$. The question is how to characterize the convergence of the net $\{x_\alpha\}_{\alpha\in A}$ in $(X,\tau_{\text{final}})$?

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It's easy to get the following characterization for net convergence in initial topology:

Given a family of functions $g_i:X\to Y_i$, let $\tau_{\text{initial}}$ be the initial topology w.r.t. $\{g_i\}_{i\in I}$, that is, it's the coarsest topology such that each $g_i:(X,\tau_{\text{initial}})\to (Y_i,\mathscr T_i)$ is continuous. Then the net $\{x_\alpha\}_{\alpha\in A}\to x$ in $(X,\tau_{\text{initial}})$ if and only if for all $i\in I$, $\{g_i(x_\alpha)\}_{\alpha\in A}\to g_i(x)$ in $(Y_i,\mathscr T_i)$.

So is there any analogous characterization for that of final topology? Any comments or hints will be appreciated!

Answer 1

There is no simple characterization of when a net converges in the final topology. In particular, arguably the simplest case is when your family just consists of a single space $Y$ with a surjective map $f:Y\to X$. In that case the final topology is just the quotient topology, but there is no simple description of when a net converges in the quotient topology in terms of convergence of nets in the original space. For examples of how some naive guesses can go wrong, you may be interested in the post Lifting a convergent net through a quotient map.

To give a bit of a broader perspective, convergence of a net in a space $X$ is equivalent to continuity of a certain map $I\to X$ for a certain space $I$ (see this answer of mine). The initial topology is a type of limit in the category of topological spaces, and maps into a limit are characterized by a universal property, and so convergence of nets in a limit space has a simple characterization. On the other hand, the final topology is a colimit, and there is no universal property for maps into a colimit (instead the universal property is for maps out of it). So it should not be surprising that there is no nice characterization of convergence of nets in a colimit.

Answer 2

A convergence definition on a set $X$ that just respects subnets i.e. if a net is defined to converge then any subnet of that also converges by the definition, generates the finest topology in which every a-priori convergent net converges. But there might be convergent nets in that generated topology that are not among the a-priori ones. The usual textbook properties of topological convergence prevent this but a topology can be generated with less than those.

If there is a topology $\tau$ on $X$ already and the a-priori convergent nets do converge in $\tau$, then the generated topology may be the same as $\tau$ and it may be said that the convergence generates $\tau$. But, as explained above, such a convergence may not fully characterize the topological convergence of $\tau$. For the quotient topology $\pi\colon Y\rightarrow X$, the convergence definition $x_\lambda\rightarrow x$ if there is a $y_\lambda\in Y$ such that $\pi(y_\lambda)=x_\lambda$ such that $y_\lambda\rightarrow y$ and $\pi(y)=x$, generates the quotient topology but there may be nets $x_\lambda\in X$ which converge but have no such convergent lift to $Y$.

It may be useful to note that you can still work with an a-priori convergence definition when it just generates a topology because it does characterize closed sets and continuity, even though you don't necessarily get all the topologically convergent nets. For example, with the quotient topology $\pi\colon Y\rightarrow X$, $f\colon X\rightarrow Z$ is continuous if and only if $f\circ\pi$ is, from which $f$ is continuous if and only if $f(x_\lambda)\rightarrow f(x)$ for all $x_\lambda$ that do lift to $Y$. You have to be careful, though: just because some topological notions are characterized by the generating nets does not mean they all are. Some very plausible looking statements involving generating nets may be false.

For a summary that tries to be self-contained and generally accessible, you might be interested in the Appendix of https://www.aimsciences.org/article/doi/10.3934/jcd.2021003.