Classification is a fundamental task in machine learning and data mining. Existing classification methods are designed to classify unknown instances within a set of previously known training classes. Such a classification takes the form of a prediction within a closed-set of classes. However, a more realistic scenario that fits real-world applications is to consider the possibility of encountering instances that do not belong to any of the training classes, $i.e.$, an open-set classification. In such situation, existing closed-set classifiers will assign a training label to these instances resulting in a misclassification. In this paper, we introduce Galaxy-X, a novel multi-class classification approach for open-set recognition problems. For each class of the training set, Galaxy-X creates a minimum bounding hyper-sphere that encompasses the distribution of the class by enclosing all of its instances. In such manner, our method is able to distinguish instances resembling previously seen classes from those that are of unknown ones. To adequately evaluate open-set classification, we introduce a novel evaluation procedure. Experimental results on benchmark datasets show the efficiency of our approach in classifying novel instances from known as well as unknown classes.