There is a locally compact Hausdorff space of weight אω which is linearly Lindelöf and not Lindelöf.

In the structure theory of C*-algebras an important role is played by certain topological spaces X which, though locally compact in a certain sense, do not in general satisfy even the weakest separation axiom. This note is concerned with the construction of a compact Hausdorff topology for the space G(X) of all closed subsets of such a space X. This construction occurs naturally in the theory o...

We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces is K4 and the modal logic of weakly scattered Stone spaces is K4G. As a corollary, we obtain that K4 is also the modal logic of compact Hausdorff spaces and K4G is the modal logic of weakly scattered compact Hausdorff spaces. §

A first-order expansion of the R-vector space structure on R does not define every compact subset of every Rn if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if A ⊆ Rk is closed and the Hausdorff dimension of A exceeds the topological dimension of A, then every compact subset of every Rn can be constructed from A using finitely many boolea...

Given a bounded metric space X, the set of closed sets of X supports a metric, the Hausdorff metric. Whether X is bounded or not, there is a compact, locally compact topology on the space of closed sets. If A,B ⊂ X are closed sets, define their Hausdorff distance dH(A,B) to be the number inf { r | B is in the r − neighborhood of A andA is in the r − neighborhood of B }. We can say this more pre...

Let R be a quasi-order on a compact Hausdorff topological space X. We prove that if X is scattered, then R satisfies the Priestley separation axiom if and only if R is closed in the product space X × X. Furthermore, if X is not scattered, then we show that there is a quasi-order on X that is closed in X × X but does not satisfy the Priestley separation axiom. As a result, we obtain a new charac...

For an algebra A of complex-valued, continuous functions on a compact Hausdorff space (X, τ), it is standard practice to assume that A separates points in the sense that for each distinct pair x, y ∈ X, there exists an f ∈ A such that f(x) 6= f(y). If A does not separate points, it is known that there exists an algebra  on a compact Hausdorff space (X̂, τ̂) that does separate points such that th...

We explore representing the compact subsets of a given represented space by infinite sequences over Plotkin's T. We show that compact Hausdorff spaces with a proper dyadic subbase admit representations of their compact subsets in such a way that compact sets are essentially underspecified points. We can even ensure that a name of an n-element compact set contains at most n occurrences of ⊥.

By the Riesz Representation Theorem for locally compact Hausdorff spaces, for every positive linear functional I on K(X) there is a measure μ such that I(f) = R f dμ, where K(X) is the set of continuous real functions with compact support on the locally compact Hausdorff space X. In this article we prove a uniformly computable version of this theorem for computably locally compact computable Ha...