  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 observer's 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.
