Bitmap mode is a color space. (or the lack of color)
Saving a bitmap file is saving a file in bitmap mode.
If you need transparent whites, you need to save a EPS bitmap file.
If you need to colorize a bitmap file to be placed into In Design or Illustrator, you need to save a TIF file in bitmap mode.
It looks as if there are (at least) three different usages for the term for bitmap that may contribute to your question. (And please someone correct me if Im wrong.):
"Bitmap" used generically to describe any pixel based graphic. (As opposed to a Vector graphic)
"Bitmap Mode" in Photoshop refers to a 1-bit mode where the pixels are either black or white.
BMP (bitmap), a standard image file format used in a Windows environment.
An ancient term from the first century of computing. Much of the early writings have been lost. Bitmap means pixels. Bitmap mode means pixels are black or white, no in-between shades of gray. The bitmap file format is common to Windows. The file format .bmp can contain pixels as shades of gray or color.
To learn the scientific parameters of the .bmp file format do a search on google.
To go a bit further, yes, TIFF is a very good format in the right circumstances, and it does support transparency. Usually you DON’T want to save as .bmp !
So, what do you want to do with the file(s)?
If he’s giving the files to a commercial printer he should choose tiff over bmp.
We’re designing tshirts using illustrator, with transparent bitmaps and color blocks showing through from underneath.
I was concerned that if the mode is bitmap, the file format should be bitmap as well. Sorta like saving a CMYK image as a JPEG – it’s possible, but incorrect.
Now I realize that it’s ok to save bitmap mode as a TIFF.
Just because you can, does not mean you should.
