/sci/ - Science & Math

For [math]{\mathcal C}[/math] any collection of sets (e.g. a topology), and [math]U\colon {\mathbb N}\to {\mathcal C}[/math] a sequence of sets, i.e. [math]U_n\in C[/math] for all [math]n[/math],
is there a name for the property that

[math] \lim_{m\to\infty} f\left(\bigcup_{n=0}^m U_n\right)[/math],
exists

and similarly is there a name for
[math]f\left(\bigcup_{n=0}^\infty U_n\right) = \lim_{m\to\infty} f\left(\bigcup_{n=0}^m U_n\right)[/math],

where as usual, the infinite union is given by
[math]x\in \bigcup_{n=0}^\infty U_n\iff \exists k. x\in U_k[/math],