Topology is the branch of mathematics that seeks to understand and describe spatial relations. A number of studies have examined the human perception of topology — in particular, whether adults and young children perceive and differentiate objects based on features like closure, boundedness, and emptiness. Topology is about more than ‘wholes and holes’, however; it also offers an efficient language for representing network structure. Topological maps, common for subway systems across the world, are an example of how effective this language can be. Inspired by this idea, here we examine ‘intuitive network topology’. We first show that people readily differentiate objects based on several different features of topological networks. We then show that people both remember and match objects in accordance with their topology, over and above substantial variation in their surface features. These results demonstrate that humans possess an intuitive understanding for the basic topological features of networks, and hint at the possibility that topology may serve as a format for representing relations in the mind.