This study introduces and investigates a new class of sets, termed nano-semi-weakly generalized closed sets (NSWG-CS), within the framework of nano-topological spaces (NTS). Their key properties, formal definitions, and relationships with other generalized closed sets are examined. To bridge theoretical insights with computational applications, we present a machine learning (ML)-based automated theorem verification system. A graph neural network model (GNN) is implemented to classify and validate the NSWG-CS by leveraging structured representations of subset relations. The model is trained on a synthetic dataset and achieves an accuracy of 86.7%, an F1-score of 85.4%, a recall of 83.2% and precision of 87.8%, demonstrating reliable and realistic performance. These findings highlight the feasibility of applying ML techniques to verify mathematical properties within nano-topological structures. The integration of nano-topology and artificial intelligence contributes to the broader field of computational mathematics and automated theorem verification.