Cleaning implements having structures for retaining a sheet

A cleaning implement for use with a sheet, comprising:

• a handle;
• a cleaning head attached to said handle;
• at least one attachment structure disposed on said cleaning head for receiving and retaining the sheet about said cleaning head, wherein said attachment structure is formed from a flexible material;
• and
• wherein said attachment structure further comprises a base triangle having an apex and a plurality of substantially pie-shaped sections whose apexes meet at a substantially common point adjacent said base triangle and whose apexes are distinct from said base triangle's apex, wherein two sides of each of said base triangle and said pie-shaped sections are defined by slits passing through said flexible material such that each of said base triangle and said pie-shaped sections can be deflected to receive the sheet and wherein each of said slits terminates at one end with a substantially circular opening.