Kelim, Chapter Eighteen, Mishnah Nine
1) A bed half of which was stolen or lost, or one which brothers or joint owners divided between themselves, becomes clean.
2) If it was restored it is susceptible to uncleanness henceforth.
3) A bed may contract uncleanness and be rendered clean when all its parts are bound together, the words of Rabbi Eliezer.
a) But the sages say: it can contract uncleanness and is rendered clean in single parts.
Section one: If a bed was half stolen or lost (perhaps he lost it under the bed?) or if it was divided up by two different owners (I would want the side closest to the TV) the bed is clean because it is unlikely that it will ever be restored to its whole state.
Section two: If it is restored, then the bed is only clean from that point and onward. The impurity it had before half of it was lost, stolen or divided does not return to it.
Section three: I am going to explain this section according to Albeck’s commentary. Others explain it differently.
According to Rabbi Eliezer, if a bed is broken into pieces and then one part of the broken piece is defiled, the whole broken piece is defiled. In other words, just as all of a bed is considered defiled if one of its pieces is defiled, so too all of a piece of a bed is considered to be defiled if any one of its components is defiled.
If one immerses an entire piece of a bed in a mikveh (a piece made up of several components) then the whole piece is pure, and the pieces do not serve as a barrier for the effectiveness of the mikveh (this is called hatzitzah in Hebrew).
The other rabbis disagree. If a bed is broken into pieces and one component becomes impure, the whole piece is not thereby impure. A piece of a bed does not have the same rules as the whole bed.
Similarly, if one immerses a piece of a bed in a mikveh, he must immerse each part separately. If he does not, each part will serve as a barrier (hatzitzah) for the part that it is connected to.