While the topology of the Universe is at present not specified by any known fundamental theory, it may in principle be determined through observations. In particular, a non-trivial topology will generate pairs of matching circles of temperature fluctuations in maps of the cosmic microwave background, the so-called circles-in-the-sky. A general search for such pairs of circles would be extremely costly and would therefore need to be confined to restricted parameter ranges. To draw quantitative conclusions from the negative results of such partial searches for the existence of circles we need a concrete theoretical framework. Here we provide such a framework by obtaining constraints on the angular parameters of these circles as a function of cosmological density parameters and the observers position. As an example of the application of our results, we consider the recent search restricted to pairs of nearly back-to-back circles with negative results. We show that assuming the Universe to be very nearly flat, with its total matter-energy density satisfying the bounds $ 0 <|Omega_0 - 1| lesssim 10^{-5}$, compatible with the predictions of typical inflationary models, this search, if confirmed, could in principle be sufficient to exclude a detectable non-trivial cosmic topology for most observers. We further relate explicitly the fraction of observers for which this result holds to the cosmological density parameters.
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