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Order intervals in C ( K ) . Compactness, coincidence of topologies, metrizability

Zbigniew Lipecki (2022)

Commentationes Mathematicae Universitatis Carolinae

Let K be a compact space and let C ( K ) be the Banach lattice of real-valued continuous functions on K . We establish eleven conditions equivalent to the strong compactness of the order interval [ 0 , x ] in C ( K ) , including the following ones: (i) { x > 0 } consists of isolated points of K ; (ii) [ 0 , x ] is pointwise compact; (iii) [ 0 , x ] is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on [ 0 , x ] ; (v) the strong and weak topologies coincide on [ 0 , x ] . Moreover, the weak topology and that of pointwise convergence...

Order-like structure of monotonically normal spaces

Scott W. Williams, Hao Xuan Zhou (1998)

Commentationes Mathematicae Universitatis Carolinae

For a compact monotonically normal space X we prove:   (1)   X has a dense set of points with a well-ordered neighborhood base (and so X is co-absolute with a compact orderable space);   (2)   each point of X has a well-ordered neighborhood π -base (answering a question of Arhangel’skii);   (3)   X is hereditarily paracompact iff X has countable tightness. In the process we introduce weak-tightness, a notion key to the results above and yielding some cardinal function results on monotonically normal...

Pairwise weakly Hausdorff spaces

M. E. El-Shafei (2005)

Archivum Mathematicum

In this paper, we introduce and investigate the notion of weakly Hausdorffness in bitopological spaces by using the convergent of nets. Several characterizations of this notion are given. Some relationships between these spaces and other spaces satisfying some separation axioms are studied.

Partial discretization of topologies

M. Bonanzinga, F. Cammaroto, M. V. Matveev (2000)

Bollettino dell'Unione Matematica Italiana

In questo lavoro daremo una construzione che aumenta il numero di sottospazi chiusi e discreti dello spazio e daremo alcune applicazioni di tale construzione.

Partitions of compact Hausdorff spaces

Gary Gruenhage (1993)

Fundamenta Mathematicae

Under the assumption that the real line cannot be covered by ω 1 -many nowhere dense sets, it is shown that (a) no Čech-complete space can be partitioned into ω 1 -many closed nowhere dense sets; (b) no Hausdorff continuum can be partitioned into ω 1 -many closed sets; and (c) no compact Hausdorff space can be partitioned into ω 1 -many closed G δ -sets.

Point-countable π-bases in first countable and similar spaces

V. V. Tkachuk (2005)

Fundamenta Mathematicae

It is a classical result of Shapirovsky that any compact space of countable tightness has a point-countable π-base. We look at general spaces with point-countable π-bases and prove, in particular, that, under the Continuum Hypothesis, any Lindelöf first countable space has a point-countable π-base. We also analyze when the function space C p ( X ) has a point-countable π -base, giving a criterion for this in terms of the topology of X when l*(X) = ω. Dealing with point-countable π-bases makes it possible...

Preservation and reflection of properties acc and hacc

Maddalena Bonanzinga (1996)

Commentationes Mathematicae Universitatis Carolinae

The aim of the paper is to study the preservation and the reflection of acc and hacc spaces under various kinds of mappings. In particular, we show that acc and hacc are not preserved by perfect mappings and that acc is not reflected by closed (nor perfect) mappings while hacc is reflected by perfect mappings.

Properties of the class of measure separable compact spaces

Mirna Džamonja, Kenneth Kunen (1995)

Fundamenta Mathematicae

We investigate properties of the class of compact spaces on which every regular Borel measure is separable. This class will be referred to as MS. We discuss some closure properties of MS, and show that some simply defined compact spaces, such as compact ordered spaces or compact scattered spaces, are in MS. Most of the basic theory for regular measures is true just in ZFC. On the other hand, the existence of a compact ordered scattered space which carries a non-separable (non-regular) Borel measure...

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