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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...

Ordered Cauchy spaces.

Kent, Darrell C., Vainio, R. (1985)

International Journal of Mathematics and Mathematical Sciences

Ordered group invariants for one-dimensional spaces

Inhyeop Yi (2001)

Fundamenta Mathematicae

We show that the Bruschlinsky group with the winding order is a homomorphism invariant for a class of one-dimensional inverse limit spaces. In particular we show that if a presentation of an inverse limit space satisfies the Simplicity Condition, then the Bruschlinsky group with the winding order of the inverse limit space is a dimension group and is a quotient of the dimension group with the standard order of the adjacency matrices associated with the presentation.

Ordered K-theoryand minimal symbolic dynamical systems

Christian Skau (2000)

Colloquium Mathematicae

Recently a new invariant of K-theoretic nature has emerged which is potentially very useful for the study of symbolic systems. We give an outline of the theory behind this invariant. Then we demonstrate the relevance and power of the invariant, focusing on the families of substitution minimal systems and Toeplitz flows.

Ordered spaces and quasi-uniformities on spaces of continuous order-preserving functions.

Koena Rufus Nailana (2000)

Extracta Mathematicae

In this paper we introduce and investigate the notions of point open order topology, compact open order topology, the order topology of quasi-uniform pointwise convergence and the order topology of quasi-uniform convergence on compacta. We consider the functorial correspondence between function spaces in the categories of topological spaces, bitopological spaces and ordered topological spaces. We obtain extensions to the topological ordered case of classical topological results on function spaces....

Ordered spaces with special bases

Harold Bennett, David Lutzer (1998)

Fundamenta Mathematicae

We study the roles played by four special types of bases (weakly uniform bases, ω-in-ω bases, open-in-finite bases, and sharp bases) in the classes of linearly ordered and generalized ordered spaces. For example, we show that a generalized ordered space has a weakly uniform base if and only if it is quasi-developable and has a G δ -diagonal, that a linearly ordered space has a point-countable base if and only if it is first-countable and has an ω-in-ω base, and that metrizability in a generalized ordered...

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...

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