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The concept of mixed multiset topology was introduced and investigated by different researchers from different aspects. In this paper, we introduce the notion of mixed multiset ideal topological space. Further, we define the concepts of τ₁(τ₂)-pre-𝓘-open mset, τ₁(τ₂)-semi-𝓘-open mset, τ₁(τ₂)-α-𝓘-open mset and τ₁(τ₂)-δ-𝓘-open mset in mixed multiset ideal topological space. We investigate on these generalized open multisets.

In this paper, we have investigated one of the basic topological properties, called Metrizability in multiset topological space. Metrizable spaces are those topological spaces which are homeomorphic to a metric space. So, we first give the notion of metric between two multi-points in a finite multiset and studied some significant properties of a multiset metric space. The notion of metrizability is then studied by using this metric. Besides, the Urysohn’s lemma which is considered to be one of the important tools in studying some metrization theorems in topology is also discussed in context with multisets.

The root open msets and root closed msets associated with an mset topology are introduced and their basic properties are discussed.

In this article based on the concept of quasi-coincidence of a multipoint and a multiset, we introduce the concept of multiset mixed topological space. We have investigated some of the properties of the introduced multiset mixed topological space.

In this paper, we show the redundancies of multiset topological spaces. It is proved that $(P^\\star(U),\\sqsubseteq)$ and $(Ds(\\varphi(U)),\\subseteq)$ are isomorphic. It follows that multiset topological spaces are superfluous and unnecessary in the theoretical view point.

In this article, we use a completely new technique to formulate the definitions of ( τ 1 , τ 2 ) ∗ -pre-open mset, ( τ 1 , τ 2 ) ∗ -semi-open mset, ( τ 1 , τ 2 ) ∗ - α -open mset and ( τ 1 , τ 2 ) ∗ - β -open mset via two multiset topologies. We prove that every τ 2 -open mset is ( τ 1 , τ 2 ) ∗ -pre-open mset, ( τ 1 , τ 2 ) ∗ -semi-open mset, ( τ 1 , τ 2 ) ∗ - α -open mset and ( τ 1 , τ 2 ) ∗ - β -open mset. Furthermore, we introduce the notions of several weaker forms of multiset continuity between two multiset topological spaces. Finally, a decomposition of these weaker types of multiset continuity has been established.

The purpose of this paper is to investigate the relationship between multiset topology ( M -topology) and generalized rough multiset induced by multiset relation. The notion of transitive closure of a relation is proposed in multiset context. The notable results we establish here are (1) any serial multiset relation induce an M -topology (2) any reflexive multiset relation R and its transitive closure t ( R ) induce the same M -topology, i.e., τ ( R ) = τ ( t ( R ) ) . The interior and closure operators of the M -topology τ ( R ) is obtained and discuss their relationship with multiset relation R , when R is a reflexive multiset relation. On the other part, we consider the case when R is a tolerance multiset relation.

The significance of connectedness and compactness plays a vital role in studying the properties of topological spaces. Numerous authors explored several characterizations of topological spaces using these notions. Researchers had attempted to generalize the concepts of connectedness and compactness in various generalized topological spaces like ideal topological spaces, soft topological spaces, fuzzy topological spaces etc. In the year 2013, Babitha and John studied the concept of connectedness in soft multi topological spaces. Since then the study of these concepts has created a gap in the literature. The uniosity about the generalizations of connectedness and compactness in soft multi topological spaces explored this article.

In this article we have introduced the notion of multiset local function on an ideal topological space using the concept of q-neighbourhood in a multiset topological space. The Kuratowski closure operator has also been extended to multiset topological space with the aim to generate new topologies as well as multiset ideal topology.

We study -valued quandles and -corack bialgebras. These structures are closely related to topological field theories in dimensions and , to the set-theoretic Yang–Baxter equation, and to the -valued groups, which have attracted considerable attention or researchers. We elaborate the basic methods of this theory, find an analogue of the so-called coset construction known in the theory of -valued groups, and construct -valued quandles using -multiquandles. In contrast to the case of -valued groups, this construction turns out to be quite rich in algebraic and topological applications. We study the properties of -corack bialgebras, which play a role similar to that of bialgebras in group theory.