If you are familiar with sober spaces, you know that an irreducible closed subset of a topological space Z<\/em> is a non-empty closed subset C<\/em> such that, for all closed subsets C<\/em>1<\/sub> and C<\/em>2<\/sub>, if C<\/em> is included in C<\/em>1<\/sub> \u222a C<\/em>2<\/sub>, then C<\/em> is included in C<\/em>1<\/sub> or in C<\/em>2<\/sub>.<\/p>\n\n\n\n In general, given any family L<\/em><\/strong> of subsets of a set Z<\/em>, let me say that E<\/em> \u2208 L<\/em><\/strong> is irreducible<\/em> if and only if E<\/em> is non-empty and for all elements E<\/em>1<\/sub> and E<\/em>2<\/sub> of L<\/em><\/strong>, if E<\/em> is included in E<\/em>1<\/sub> \u222a E<\/em>2<\/sub>, then E<\/em> is included in E<\/em>1<\/sub> or in E<\/em>2<\/sub>.<\/p>\n\n\n\n Let me call a family L<\/em><\/strong> of subsets of Z<\/em> irredundant<\/em> if and only if all its non-empty elements are irreducible. More specifically, I will be interested in irredundant \u2229-semilattices<\/em>, where a \u2229-semilattice is simply a collection of sets that is closed under binary intersections. That may look like a stupidly overconstrained notion, to the point that one may legitimitely ask whether there is any non-trivial example.<\/p>\n\n\n\n This is a very interesting notion, as I will attempt to demonstrate. It was introduced in [1], and the credit is entirely due to the late Klaus Keimel. Also, it is at the root of a clever argument due to Zhenchao Lyu and Xiadong Jia [2], which exactly characterizes when the Smyth powerdomain Q<\/strong>(X<\/em>) of a topological space X<\/em> is core-compact.<\/p>\n\n\n\n Two notes, before I start:<\/p>\n\n\n\n There are at least two examples of an irredundant \u2229-semilattice in [1], and here is the one which will be the center of our attention in this post.<\/p>\n\n\n\n Fact.<\/strong> For every topological space X<\/em>, the base of open sets \u2610U<\/em> of the Smyth hyperspace Q<\/strong>(X<\/em>), where U<\/em> ranges over the open subsets of X<\/em>, is an irredundant \u2229-semilattice.<\/p>\n\n\n\n Let me recall that Q<\/strong>(X<\/em>) is the set of non-empty compact saturated subsets of X<\/em>. (Omitting “non-empty” would not change much in the sequel.) The collection of sets \u2610U<\/em>, U<\/em> \u2208 O<\/strong>X<\/em>, is a base for a topology called the upper Vietoris topology<\/em>, and Q<\/strong>(X<\/em>) with that topology is the Smyth hyperspace<\/em> of X<\/em>.<\/p>\n\n\n\n The proof of the above Fact is easy. The sets \u2610U<\/em> form a \u2229-semilattice because \u2610U<\/em> \u2229 \u2610V<\/em> = \u2610(U<\/em> \u2229 V<\/em>). Let now \u2610U<\/em> be non-empty and included in \u2610V<\/em>1<\/sub> \u222a \u2610V<\/em>2<\/sub>. For the sake of contradiction, let us assume that \u2610U<\/em> is not included in \u2610V<\/em>1<\/sub>, and not included in \u2610V<\/em>2<\/sub> either. Then there is a non-empty compact saturated subset Q<\/em>1<\/sub> of X<\/em> that is included in U<\/em> but not in V<\/em>1<\/sub>, and there is a non-empty compact saturated subset Q<\/em>2<\/sub> of X<\/em> that is included in U<\/em> but not in V<\/em>2<\/sub>. We form Q<\/em>1<\/sub> \u222a Q<\/em>2<\/sub>, which is a non-empty compact saturated subset of X<\/em> that is included in U<\/em>, but is included neither in V<\/em>1<\/sub> nor in V<\/em>2<\/sub>, contradicting \u2610U<\/em> \u2286 \u2610V<\/em>1<\/sub> \u222a \u2610V<\/em>2<\/sub>.<\/p>\n\n\n\n The other example is the collection of crescents of the form \u2610U<\/em>\u2013\u2662V<\/em> in the space of compact lenses (also known as the Plotkin powerdomain) with the Vietoris topology. I will not expand on that.<\/p>\n\n\n\n First, let me write \u2191Q<\/sub><\/strong> Q<\/em> for the the upward closure of a single point Q<\/em> of Q<\/strong>(X<\/em>). That is taken with respect to the specialization ordering of Q<\/strong>(X<\/em>), which is reverse<\/em> inclusion \u2287. Hence \u2191Q<\/sub><\/strong> Q<\/em> is the set of non-empty compact saturated sets Q’<\/em> that are included in<\/em> Q<\/em>. I know, this may sound confusing at first.<\/p>\n\n\n\n Extending the notation \u2610U<\/em>, I could have just written \u2191Q<\/sub><\/strong> Q<\/em> as \u2610Q<\/em>, the set of non-empty compact saturated sets included in Q<\/em>. The \u2610 operator, defined by \u2610A<\/em> \u225d {Q<\/em> \u2208 Q<\/strong>(X<\/em>) | Q<\/em> \u2286 A<\/em>} for every subset A<\/em> of X<\/em>, is monotonic and commutes with finite intersections.<\/p>\n\n\n\n It is well-known that, for every locally compact space X<\/em>, Q<\/strong>(X<\/em>) is locally compact. Indeed, let Q<\/em> \u2208 Q<\/strong>(X<\/em>), and let U<\/em><\/strong> be an open neighborhood of Q<\/em>. U<\/em><\/strong> contains a basic open set \u2610U<\/em>, where U<\/em> is open in X<\/em>, such that Q<\/em> is in \u2610U<\/em>, namely such that Q<\/em> is included in U<\/em>. By the interpolation property in locally compact spaces (Proposition 4.8.14 in the book<\/a>), there is (necessarily non-empty) compact saturated subset Q’<\/em> of X<\/em> such that Q<\/em> \u2286 U’<\/em> \u2286 Q’<\/em> \u2286 U<\/em>, where U’<\/em> is the interior of Q’<\/em>. Therefore Q<\/em> \u2208 \u2610U<\/em>‘ \u2286 \u2610Q’<\/em> \u2286 \u2610U<\/em>. Since \u2191Q<\/sub><\/strong> Q<\/em>‘ = \u2610Q’<\/em>, we can interpret the latter as saying that Q<\/em> is in the interior of \u2191Q<\/sub><\/strong> Q<\/em>‘, which is itself included in \u2610U<\/em>, hence in U<\/em><\/strong>.<\/p>\n\n\n\n\n
The main example: the irredundant base of the upper Vietoris topology on Q<\/strong>(X<\/em>)<\/h2>\n\n\n\n
When is Q<\/strong>(X<\/em>) core-compact? The Lyu-Jia theorem<\/h2>\n\n\n\n