Universally Kuratowski–Ulam spaces

Abstract

We introduce the notions of Kuratowski-Ulam pairs of topological spaces and universally Kuratowski-Ulam space. A pair(X,Y) of topological spaces is called a Kuratowski-Ulam pair if the Kuratowski-Ulam Theorem holds inX◊Y. A spaceY is called a universally Kuratowski-Ulam (uK-U) space if (X,Y ) is a Kuratowski-Ulam pair for every space X. Obviously, every meager in itself space is uK-U. Moreover, it is known that every space with a countable -basis is uK-U. We prove the following: • every dyadic space (in fact, any continuous image of any product of separable metrizable spaces) is uK-U (so there are uK-U Baire spaces which do not have countable -bases); • every Baire uK-U space is ccc. 1. Kuratowski-Ulam pairs. We use standard set-theoretical notions. In particular, ordinal numbers will be identified with the set of their predeces- sors and cardinal numbers with the initial ordinals. For a set A and a cardinal , (A) < is the family of all subsets of A with cardinality less than . Similarly we define the families (A) and (A) . The symbols X, Y , Z denote topological spaces, M(X) denotes the family of all meager subsets inX. ForE X◊Y andx 2 X,Ex denotes thex-section of E, etc. A family U of non-empty open subsets of X is called a pseudo-basis ( -basis for short) of X if every non-empty open set W in X contains a U 2 U. A topological space X is -cc if there is no family of size of