Last time, we had stated and proved the Dolecki-Greco-Lechicki theorem: every regular Čech-complete space is consonant. I would like to show that there are some other classes of consonant spaces, among T0 spaces. The results are going to be easy consequences of results from the book.
We start with the following.
Theorem. Every Gδ subspace of a well-filtered locally compact space is consonant.
Recall that a space is well-filtered if and only if, for every filtered family of compact saturated sets whose intersection is contained in some open set U, one member of the family must be included in U (Section 8.3.1 in the book). Every sober space, notably, is well-filtered. For a T0 locally compact space, in fact, sobriety and well-filteredness are equivalent properties (Propositions 8.3.5 and 8.3.8 in the book).
Before we start the proof, let us recall that every locally compact space is consonant. The value of the theorem lies in the fact that Gδ subspaces of consonant spaces need not be consonant [1, Proposition 7.3].
Proof. Let Y be a Gδ subset of X, where X is well-filtered and locally compact. We can write V as the intersection of an antitone sequence of open subsets Vn of X. By antitone, I mean that V0 ⊇ V1 ⊇ … ⊇ Vn ⊇ … Let U be a Scott-open family of open subsets of Y, and U be in U.
Write int for the interior operator in X. By the definition of the subspace topology, there is an open subset U’ of X such that U’ ⋂ Y = U. By local compactness, U’ ⋂ V0 is the union of the directed family of sets int(Q), where Q ranges over the family Q0 of compact saturated subsets of U’ ⋂ V0. (That family is clearly directed. Moreover, local compactness tells us that every point of U’ ⋂ V0 lies in int(Q) for some Q in Q0.) The union over all Q in Q0 of the sets int(Q) ⋂ Y is then equal to U’ ⋂ V0 ⋂ Y, that is, to U ⋂ V0, i.e., U.
Since U is in U and U is Scott-open, int(Q) ⋂ Y is in U for some Q in Q0. Let Q0 be this compact saturated set Q, U’0 be int(Q0), and U0 be equal to U’0 ⋂ Y. Note that U0 is in U, and that U’0 ⊆ Q0 ⊆ U’ ⋂ V0.
Do the same thing with U’0 ⋂ V1 instead of U’ ⋂ V0. There is a compact saturated subset Q1 of U’0 ⋂ V1 such that int(Q1) ⋂ Y is in U. Let U’1 be int(Q1), and U1 be equal to U’1 ⋂ Y. Note that U1 is in U, and that U’1 ⊆ Q1 ⊆ U’0 ⋂ V1.
Iterating this construction, we obtain for each natural number n a compact saturated subset Qn of X, an open subset U’n of X, and an open subset Un of Y, such that Un is in U, and U’n+1 ⊆ Qn+1 ⊆ U’n ⋂ Vn, for every n in N.
Let Q be the intersection of all Qn, n in N. Since X is well-filtered, Q is compact saturated in X (Proposition 8.3.6 in the book).
Since every Qn is included in Vn, Q is included in the intersection of the Vns, which happens to be Y. We check that Q is also compact saturated in Y. The specialization quasi-ordering of Y is the restriction of that of X (Proposition 4.9.5 in the book), so Q is upwards-closed, namely, saturated, in Y. For every open cover (Wj)j ∈ J of Q in Y, we write each Wj as the intersection of some open subset W’j of X with Y. Then (W’j)j ∈ J is an open cover of Q in X, from which we can extract a finite subcover (W’j)j ∈ K (K finite). It is then clear that (Wj)j ∈ K is a finite subcover of Q, since Q is included in Y.
We observe that Q is included in Q0, which is included in U’ ⋂ V0,. Since it is also included in Y, it is included in U’ ⋂ Y = U. Therefore U is in ￭Q.
It remains to show that every W in ￭Q is in U. Write W as the intersection of some open subset W’ of X with Y. Since Q, which is equal to the filtered intersection of the compact saturated subsets Qn, is included in W, hence in W’, some Qn is included in W’ by well-filteredness. It follows that U’n is included in W’. Taking intersections with Y, Un is included in W. Since Un is in U, so is W. ☐
That theorem has the following nice consequence.
Corollary. Every continuous Yoneda-complete quasi-metric space X, d is consonant in its d-Scott topology.
Proof. Every continuous dcpo is sober in its Scott topology (Proposition 8.2.12 (b) in the book), hence well-filtered, and also locally compact (Corollary 5.1.36). The definition of the d-Scott topology means that the map x ⟼ (x, 0) is a topological embedding of X into its dcpo of formal balls B(X, d), and we equate X with a subspace of B(X, d) through this map.
For every ε > 0, let Vε be the set of formal balls (x, r) whose radius is < ε. This is Scott-open in B(X, d): by the Kostanek-Waszkewicz theorem (Theorem 7.4.27), the supremum of a directed family of formal balls (xi, ri) is of the form (x, r) where r = inf ri (and x is the d-limit of the net consisting of the points xi); if r < ε, then some ri is also < ε.
The family of open sets V1/2n, n in N, is then an antitone sequence of open sets of B(X, d), whose intersection equals X. In other words, X is a Gδ subspace of B(X, d), and we conclude by applying the previous Theorem. ☐
We retrieve, notably, that every complete metric space is consonant, because every complete metric space is continuous Yoneda-complete, and its open ball topology coincides with the d-Scott topology. This yields another proof of that result, which is classically obtained by noticing that every complete metric space is (completely) regular and Čech-complete.
What we gain is that the above corollary applies to a whole family of T0 spaces that are far from being regular: recall that a T0 regular space is automatically T2, whereas most continuous Yoneda-complete quasi-metric spaces are not.
- Dolecki, Szymon, Greco, Gabriele H., and Lechicki, Alojzy, 1995. When Do the Upper Kuratowski Topology (Homeomorphically, Scott Topology) and the Co-Compact Topology Coincide? Transactions of the American Mathematical Society, 347(8), 2869–2884.