Plithogenic Topological Spaces

Plithogenic sets enrich multi-attribute appurtenance degrees with contradiction degrees measured relative to dominant attribute values. We investigate how a topology can be defined directly on scalar plithogenic profiles. The most immediate construction—using the native plithogenic AND and OR as topological intersection and union—has a structural obstruction: in the minimum/maximum specialization, the binary operators are non-associative for every contradiction level 0 < c < 1, while the natural arbitrary-arity infimum/supremum blend is neither an order-theoretic meet/join nor stable under flattening of indexed families. We therefore separate topology from aggregation. On a fixed plithogenic frame, all appurtenance profiles form a complete pointwise lattice; its suprema and infima provide the genuine topological joins and meets. The original contradiction-weighted plithogenic operators are retained as additional algebraic operations, and a topology closed under them is called aggregation-compatible. We prove existence of generated compatible topologies, give criteria that force a compatible hull to be finite or infinite, establish a basis criterion, closedset duality, Kuratowski-type interior and closure properties, subspace and continuity theorems, profile-based and restricted-profile separation axioms, and preservation of threshold compactness under continuous surjections. We also show that compactness at two distinct thresholds is not comparable in general, explain why p-disjointness is unsuitable for Hausdorff separation, and give a finite example in which compatibility forces a crisp topology to acquire infinitely many graded open profiles. Earlier plithogenic hybrid topologies are distinguished explicitly, so the novelty claim is restricted to this direct fixed-frame scalar setting and to the min/max operator obstruction analyzed here.