Note that we can consider multiple sequences at the same time by using different variables; e.g. {\displaystyle (b_{n})_{n\in \mathbb {N} }} could be a different sequence than {\displaystyle (a_{n})_{n\in \mathbb {N} }} . We can even consider a sequence of sequences: {\displaystyle ((a_{m,n})_{n\in \mathbb {N} })_{m\in \mathbb {N} }} denotes a sequence whose mth term is the sequence {\displaystyle (a_{m,n})_{n\in \mathbb {N} }} .

is defined as the set of all sequences {\displaystyle (x_{i})_{i\in \mathbb {N} }} such that for each i, {\displaystyle x_{i}} is an element of {\displaystyle X_{i}} . The canonical projections are the maps pi : X → Xi defined by the equation {\displaystyle p_{i}((x_{j})_{j\in \mathbb {N} })=x_{i}} . Then the product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections pi are continuous. The product topology is sometimes called the Tychonoff topology.
I am using Windows 10 and Lightroom Classic CC.  Unlike many professional photographers who go on very specific shoots, when I travel, especially, I am shooting a wide variety of shots and in many different places.  Thus, I find it most convenient to import my photos into Lightroom with only the most basic information -- metadata, and general keywords.  Then when I have the time, I go through and sort the photos into subfolders, give them titles and more specific keywords.  I also want to number these photos.  Often, however, I will want to add more photos to a subfolder -- say my husband's photos taken at the same place.  In Photoshop, these photos can be numbered so that the numbering sequence follows the photos in the same folder that have already been numbered.

Yes, I have used this system in a multi-user setting. As noted, the key is to commit the record immediately after generating the sequence. However, if the application is one where there is very heavy transaction processing. In other words dozens of users creating records simultaneously, you might want to guard further against duplication. At the speeds computers process, it is not impossible that multiple users will grab the max value before it can be incremented and saved.

LION also carries a heavy-duty, 6 wheel automatic numbering machine with rubber faced wheels. The rubber wheels work great for metal marking and plastic marking when used with LION fast dry ink. As with the other LION numbering machines, this machine is made in Japan with precision crafted one-piece hardened steel frame with all metal interior construction. LION machines will provide years of reliable use. Ideal for sequential numbering operations to use as a date and number stamp, serial number stamp, an inspection stamp and etc. sequential numbering in publisher