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We consider topological four-dimensional ͑4D͒ gravity with an independent spin connection by using a superspace formalism. This gives rise to the basic fields of the quantized theory as well as to two pairs of extra fields, which are needed to close the BRST-anti-BRST algebra off shell. Therefore we build a gauge-fixing action written in BRST-anti-BRST exact form leading to an effective one, which allows us to fix all of the symmetries at once. In particular, the topological symmetries are fixed as in the model of topological 4D self-dual gravity. We construct the observables related to both BRST symmetry and anti-BRST symmetry. We find that the anti-BRST invariant observables are not fundamentally different from the BRST invariant ones, since there is a complete mirror symmetry between them. The obtained observables extend those constructed within the equivariant method in the context of topological 4D self-dual gravity. ͓S0556-2821͑98͒00712-7͔PACS number͑s͒: 04.50.ϩh Topological field theories provide quantum field theoretic models which incorporate a way of constructing topological invariants, for a review see Ref.͓1͔. An important example is given by Witten's topological Yang-Mills theory ͑TYMT͒ ͓2-5͔. This suggests that one can also describe the global structure of 4D manifolds by constructing topological 4D gravity as a gravitational counterpart of TYMT. In this context, various models were proposed. For example, topological 4D conformal gravity was first discussed in Ref. ͓6͔. This was developed further in Ref. ͓7͔, where the authors start with the topological action constructed from the Weyl tensor.Other examples of topological 4D gravity were considered in Refs. ͓8-10͔. There, the theory is constructed from topological combinations of the curvature tensor. In particular, in Ref. ͓10͔ the topological symmetries are fixed by anti-selfduality on both the curvature and the torsion; and the observables are derived studying the equivariant cohomology ͑see also Ref. ͓9͔͒ as in the case of TYMT ͓11͔. These observables are constructed from operators, which are only 4D forms, contrary to what happens in TYMT where the operators are forms of any degree. Let us note that it is in Ref. ͓12͔ where one has shown how to build observables from operators, which are forms of any degree, in the context of the metric approach to topological gravity.The purpose of this paper is to present topological 4D gravity with an independent spin connection in terms of a superspace formalism. The basic fields of the theory are introduced through a superconnection and its associated supercurvature. The use of the superspace formalism naturally yields the off-shell nilpotent Becchi-Rouet-Stora-Tyutin ͑BRST͒ and anti-BRST transformations. Hence it should be possible to consider the BRST and anti-BRST invariant quantum action for topological 4D gravity. At this point, we remark that the coexistence of both BRST symmetry and anti-BRST symmetry was already realized in TYMT ͓13͔ and in topological antisymmetric tensor gauge theory ͓14͔, so-cal...

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