In congested electromagnetic environments, cognitive radios require knowledge about other emitters in order to optimize their dynamic spectrum access strategy. Deep learning classification algorithms have been used to recognize the wireless signal standards of emitters with high accuracy, but are limited to classifying signal classes that appear in their training set. This diminishes the performance of deep learning classifiers deployed in the field because they cannot accurately identify signals from classes outside of the training set. In this paper, a convolution neural network based open set classifier is proposed with the ability to detect if signals are not from known classes by thresholding the output sigmoid activation. The open set classifier was trained on 4G LTE, 5G NR, IEEE 802.11ax, Bluetooth Low Energy 5.0, and Narrowband Internet-of-Things signals impaired with Rayleigh or Rician fading, AWGN, frequency offsets, and in-phase/quadrature imbalances. Then, the classifier was tested on OFDM, SC-FDMA, SC, AM, and FM signals, which did not appear in the training set classes. The closed set classifier achieves an average accuracy of 94.5% for known signals with SNRs greater than 0 dB, but by design, has a 0% accuracy detecting signals from unknown classes. On the other hand, the open set classifier retains an 86% accuracy for known signal classes, but can detect 95.5% of signals from unknown classes with SNRs greater than 0 dB.