During the spring of 2005 I kept hearing the song "Incomplete" on the radio. Anyone who knows about metric spaces should agree right away that the title suggests the song is ripe for a mathematical parody. Read below while watching the video to the original song on You Tube. Over time, posts of that video on You Tube are taken down, so if the previous link doesn't work let me know. (The lyrics to the original song, without the music, can be found here.) I wrote to one of the singers in the Klein four group to see if they were interested in recording my version. Alas, they only perform their own original songs. If you are inspired to make a video of yourself or your friends singing my parody, please send it to me. An instrumental version of the song, if you need it for background, is here. The link currently works, but if you find it is taken down then let me know.

Incomplete (Parody of "Incomplete" by the Backstreet Boys)

Empty spaces fill me up with holes
Distant limits with no place here to go
Without them within me I fail Cauchy's test
Where I converge is anybody's guess

I've tried to modify my given metric
Just to be what an analyst will seek
Frechet found the necessary notion
But without it all I'm going to be is incomplete

Interest in me is so often rare
But I am hoping for an analogue of Baire
Maybe, oh maybe
Contractions on my space
Fix just one point if you catch a lucky break

I've tried to modify my given metric
Just to be what an analyst will seek
Frechet found the necessary notion
But without it all I'm going to be is incomplete

I don't need to be compact, or even bounded totally
I just wanna have a useful property
I wanna be complete (oh please)

I've tried to modify my given metric
Just to be what an analyst will seek
Frechet found the necessary notion
But without it all I'm going to be is incomplete

Incomplete

Note: Frechet defined metric spaces and completeness in his thesis (1906). Hausdorff (1914) proved every metric space has a completion.