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Energy consumption in wireless sensor networks is an important topic, given the scarcity of resources of the node platforms. This work analyzes the relation between the increase in lifetime and the connectivity properties of a new topology control mechanism for wireless sensor networks. It is possible to extend the functional lifetime of a network by controlling its topology through a scheme that sets nodes to sleep, though at the expense of performance loss in other metrics. But even when controlling its topology, for a network to work it should remain connected at all times. Given the base network lifetime and its desired extension factor, this paper provides formulae for the required density of nodes to be deployed in order to keep the network connected with a given probability. The network connectivity is analyzed under Boolean and lognormal shadowing models for the communication channel and stochastic simulations corroborate the derived formulae. Using the proposed topology control mechanism and by adjusting the density of the node deployment, the lifetime of a network can be extended in a simple, autonomous, and scalable way.