From Freyd's covering theorem for Grothendieck topoi, it immediately follows that every Grothendieck topos E admits a hyperconnected geometric morphism 3 -C, where 7 is an etendue of (discrete) G-sheaves. As a corollary, we obtain that E admits an open surjection from a localic topos. A fundamental representation theorem in topos theory is provided by the following result of P. Freyd. If E is a Grothendieck topos, there is a connected atomic geometric morphism 7 -f &, where 7 is localic over a topos C(G) of continuous G-sets, G being a topological group. In fact, G may be chosen to be either Go, the group of permutations of N, or G1, the group of order-preserving permutations of Q. (See [1] or the excellent survey article [2].) In [4], it was shown that given 6, there is a hyperconnected geometric morphism 7 6, where 7 is an etendue, i.e., there is a surjective local homeomorphism F/X with F/X localic. Using Freyd's result, we not only obtain this theorem directly (with . a particularly nice etendue), but the fact that & has a localic cover via an open surjection [3, p. 61] also follows. To fix some notation, sh(G; B) denotes the topos of (discrete) G-sheaves, where B is a locale on which the group G acts. sh(G; B) is a basic example of etendue and is locally sh(B). Also, SG denotes the topos of G-sets. There is a canonical map SG -A C(G) [4]. THEOREM. Let E be a Grothendieck topos. There is a hyperconnected geometric h morphism 7 -h 6, where 1 is an e'tendue of G-sheaves, sh(G; B). (G can be taken to be either the group Go or Gi.) PROOF. Let C(G)[A] I+ & be connected and atomic, where C(G)[A] is the topos of C(G)-sheaves on an internal locale A in C(G). Let B be the Macneille completion of A. Thus A is the cocontinuous reflection of B along the hyperconnected geometric morphism SG -9 C(G). The following diagram can be seen to be a pullback. sh(G; B) 9 ) (G) [A] 1 l~~~~k Sc G C(G) Since pullbacks perserve hyperconnectedness [3, p. 54], 9 is hyperconnected. Connected and atomic implies hyperconnected [2, Lemma 2.1], thus the composite f sh(G; B) J9 C(G)[A] f E is hyperconnected. Take this to be h. O Received by the editors January 31, 1986. 1980 Mathematics S*ject ai&fication (1985 Revison). Primary 18B25.