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We show that in the Maxwell-Chern-Simons theory of topologically massiveelectrodynamics the Dirac string of a monopole becomes a cone in anti-de Sitterspace with the opening angle of the cone determined by the topological masswhich in turn is related to the square root of the cosmological constant. Thisproves to be an example of a physical system, {\it a priory} completelyunrelated to gravity, which nevertheless requires curved spacetime for its veryexistence. We extend this result to topologically massive gravity coupled totopologically massive electrodynamics in the framework of the theory of Deser,Jackiw and Templeton. These are homogeneous spaces with conical deficit. PureEinstein gravity coupled to Maxwell-Chern-Simons field does not admit such amonopole solution

Topologically massive magnetic monopoles