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L J -spaces

Yin-Zhu Gao (2007)

Czechoslovak Mathematical Journal

In this paper L J -spaces are introduced and studied. They are a common generalization of Lindelöf spaces and J -spaces researched by E. Michael. A space X is called an L J -space if, whenever { A , B } is a closed cover of X with A B compact, then A or B is Lindelöf. Semi-strong L J -spaces and strong L J -spaces are also defined and investigated. It is demonstrated that the three spaces are different and have interesting properties and behaviors.

Large cardinals and Dowker products

Chris Good (1994)

Commentationes Mathematicae Universitatis Carolinae

We prove that if there is a model of set-theory which contains no first countable, locally compact, scattered, countably paracompact space X , whose Tychonoff square is a Dowker space, then there is an inner model which contains a measurable cardinal.

Lindelöf indestructibility, topological games and selection principles

Marion Scheepers, Franklin D. Tall (2010)

Fundamenta Mathematicae

Arhangel’skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most 2 . Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are G δ has been more elusive. In this paper we continue the agenda started by the second author, [Topology Appl. 63 (1995)], of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinite games and selection principles, especially the Rothberger property,...

Lindelöf property and the iterated continuous function spaces

G. Sokolov (1993)

Fundamenta Mathematicae

We give an example of a compact space X whose iterated continuous function spaces C p ( X ) , C p C p ( X ) , . . . are Lindelöf, but X is not a Corson compactum. This solves a problem of Gul’ko (Problem 1052 in [11]). We also provide a theorem concerning the Lindelöf property in the function spaces C p ( X ) on compact scattered spaces with the ω 1 th derived set empty, improving some earlier results of Pol [12] in this direction.

Local/global uniform approximation of real-valued continuous functions

Anthony W. Hager (2011)

Commentationes Mathematicae Universitatis Carolinae

For a Tychonoff space X , C ( X ) is the lattice-ordered group ( l -group) of real-valued continuous functions on X , and C * ( X ) is the sub- l -group of bounded functions. A property that X might have is (AP) whenever G is a divisible sub- l -group of C * ( X ) , containing the constant function 1, and separating points from closed sets in X , then any function in C ( X ) can be approximated uniformly over X by functions which are locally in G . The vector lattice version of the Stone-Weierstrass Theorem is more-or-less equivalent...

Locally compact linearly Lindelöf spaces

Kenneth Kunen (2002)

Commentationes Mathematicae Universitatis Carolinae

There is a locally compact Hausdorff space which is linearly Lindelöf and not Lindelöf. This answers a question of Arhangel'skii and Buzyakova.

Locally compact perfectly normal spaces may all be paracompact

Paul B. Larson, Franklin D. Tall (2010)

Fundamenta Mathematicae

We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, this answers a question of S. Watson. Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space...

LΣ(≤ ω)-spaces and spaces of continuous functions

Israel Lara, Oleg Okunev (2010)

Open Mathematics

We present a few results and problems related to spaces of continuous functions with the topology of pointwise convergence and the classes of LΣ(≤ ω)-spaces; in particular, we prove that every Gul’ko compact space of cardinality less or equal to 𝔠 is an LΣ(≤ ω)-space.

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