On Multiset Topologies

Abstract

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.