The world is urbanizing quickly with nearly 4 billion people presently living in urban areas, about 1 billion of them in slums. Achieving sustainable development from rapid urbanization relies critically on creating cities without slums. We show that it is possible to diagnose systematically the central physical problem of slums—the lack of spatial accesses and related services—using a topological analysis of neighborhood maps and resolved by finding solutions to a sequence of constrained optimization problems. We set up the problem by showing that the built environment of any city can be decomposed into two types of networked spaces—accesses and places—and prove that these spaces display universal topological characteristics. We then show that while the neighborhoods of developed cities express the same common topology, urban slums fall into a different topological class. We demonstrate that it is always possible to find solutions that grow a street network in existing slums, providing universal accesses at minimal disruption and cost. We then show how elaborations of this procedure that include local preferences and reduce travel distances between places result from additional access construction. These methods are presently taking effect in neighborhoods in Cape Town (South Africa) and Mumbai (India), demonstrating their practical feasibility and emphasizing their role as a platform to enable communities and local governments to combine technical knowledge with local aspirations into contextually appropriate urban sustainable development solutions.