I am currently running mac OS Catalina 10.15.3 and I want to update to Big Sur, however, it needs a lot of space and I am actually limited on space as I have a 128 GBs laptop (big mistake, I know), my question is: will installing this new update really take as much space as indicated (for downloading I need to free up around 15 GBs then for installing the download file I need like 20-25 GBs more), will this space be forever occupied or will the old files of Catalina be deleted and I will have relatively the same amount of space before and after updating
When you update the OS from one major version to the next, many of the files will be over-written by new files with the same names. Some may be deleted.
Each new OS should take roughly the same amount of space, though some may vary. Catalina didn't have 32-bit libraries, so may be smaller than Mojave; Big Sur has Universal binaries for Intel and Apple Silicon, so may be a little bit bigger.
But essentially, the amount of space should be similar. Upgrading from an earlier OS doesn't take up more space than a new install. (Apart from all your stuff, of course.)
Big Sur is 12 Gb download, according to the Apple Store. It appears that it requests around 35 Gb of free space in addition to this at installation. You should normally (i.e. all the time) keep 10-20% of the disk as free space, to allow for temporary file writing, and SSD self-management.