Suppose I own shares of a mutual fund managed by, for example, Vanguard Investments. Now suppose, for the sake of discussion, that Vanguard decides to close its doors forever, either because they went bankrupt, or they just got tired of being in the investment business.
What becomes of my investment? This is not the same as owning shares of Vanguard stock, whose value would fall to zero when the company folded. So what happens?
The assets of the fund still exist, and you still own a share in them. Almost certainly another manager would be appointed, and almost certainly the instruments constituting the fund give the unitholders the right to do this. In the circumstances the new manager might be appointed purely to negotiate a merger with another fund, or to wind up the fund and distribute the proceeds to unitholders.
In practice, unless the manager is insolvent, it doesn’t resign without itself arranging the appointment of a successor manager, if only because that way it avoids getting sued.