The topological optimization of a conducting solid simultaneously cooled by (i) conduction to a stationary, radiatively nonparticipating fluid and (ii) surface-to-surface radiation exchange is performed to minimize the overall thermal resistance of the solid configuration. A novel dual solid method (DSM) that utilizes concurrent discrete and continuous descriptions of the solid-phase distribution is introduced. Corresponding discrete and continuous solid models are used to (i) quantify the conduction and radiation heat transfer and (ii) power a density-based topology optimization, respectively. The discrete and continuous models of the DSM are linked by sharing information pertaining to the radiation exchange process. The DSM is the first design method to incorporate the effects of surface-to-surface radiation exchange into the topological optimization of a conducting solid. The influence of the relative strengths of conduction and radiation is illustrated by performing parametric simulations involving various domain boundary temperatures and solid-phase thermal conductivities. In general, use of the DSM to account for radiation heat transfer leads to solid shapes with lower overall thermal resistances and reduced complexity, relative to shapes predicted when radiation is neglected. For the problem considered here, the DSM produces solid shapes that have overall thermal resistances up to 25% smaller relative to overall thermal resistances of shapes determined by topology optimization considering conduction processes only.