Recent observational studies of $ω$ Centauri by {\it Hubble Space Telescope} have discovered a double main sequence in the color magnitude diagrams (CMDs) of its stellar populations. The stellar population with the blue main sequence (bMS) is observationally suggested to have a helium abundance much larger, by $Δ Y∼ 0.12$, than that of the red main sequence (rMS). By using somewhat idealized models in which stars of the bMS are formed from gas ejected from those of the rMS, we quantitatively investigate whether the helium overabundance of the bMS can result from self-enrichment from massive AGB stars, from mass loss of very massive young stars, or from type II supernovae within $ω$ Cen. We show that as long as the helium enrichment is due to ejecta from the rMS formed earlier than the bMS, none of the above three enrichment scenarios can explain the observed properties of the bMS self-consistently for reasonable IMFs. The common, serious problem in all cases is that the observed number fraction of the bMS can not be explained without assuming unusually top-heavy IMFs. This failure of the self-enrichment scenarios implies that most of the helium-enriched gas necessary for the formation of the bMS originated from other external sources. We thus suggest a new scenario that most of the second generation of stars (i.e., the bMS) in $ω$ Cen could be formed from gas ejected from field stellar populations that surrounded $ω$ Cen when it was a nucleus of an ancient dwarf galaxy. Comment: 7 pages, 3 figures, accepted by ApJL