On Multiset Topologies

In this paper an attempt is made to extend the concept of topological spaces in the context of multisets (mset, for short). The paper begins with basic definitions and operations on msets. The mset space [X]w is the collection of msets whose elements are from X such that no element in the mset occurs more than finite number (w) of times. Different types of collections of msets such as power msets, power whole msets and power full msets which are submsets of the mset space and operations under such collections are defined. The notion of M-topological space and the concept of open msets are introduced. More precisely, an M-topology is defined as a set of msets as points. Furthermore the notions of basis, sub basis, closed sets, closure and interior in topological spaces are extended to M-topological spaces and many related theorems have been proved. The paper concludes with the definition of continuous mset functions and related properties, in particular the comparison of discrete topology and discrete M-topology are established.