The identification of autistic individuals using resting state functional connectivity networks can provide an objective diagnostic method for autism spectrum disorder (ASD). The present state-of-the-art machine learning model using deep learning has a classification accuracy of 70.2% on the ABIDE (Autism Brain Imaging Data Exchange) data set. In this paper, we explore the utility of topological features in the classification of ASD versus typically developing control subjects. These topological features have been shown to provide a complementary source of discriminative information in applications such as 2D object classification and social network analysis. We evaluate the performance of three different representations of topological features - persistence diagrams, persistence images, and persistence landscapes - for autism classification using neural networks, support vector machines and random forests. We also propose a hybrid approach of augmenting topological features with functional correlations, which typically outperforms the models that use functional correlations alone. With this approach, even with a simple 3-layer neural network, we are able to achieve a classification accuracy of 69.2% on the ABIDE data set. However, our experiments also show that the improvement due to topological features is not always statistically significant. Therefore, we offer a cautionary tale to the practitioners regarding the limited discriminative power of topological features derived from fMRI data for the classification of autism.