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The product of two ordinals is hereditarily dually discrete

M.Á. Gaspar-Arreola, F. Hernández-Hernández (2012)

Commentationes Mathematicae Universitatis Carolinae

In Dually discrete spaces, Topology Appl. 155 (2008), 1420–1425, Alas et. al. proved that ordinals are hereditarily dually discrete and asked whether the product of two ordinals has the same property. In Products of certain dually discrete spaces, Topology Appl. 156 (2009), 2832–2837, Peng proved a number of partial results and left open the question of whether the product of two stationary subsets of ω 1 is dually discrete. We answer the first question affirmatively and as a consequence also give...

The quasi Isbell topology on function spaces

D. N. Georgiou, A. C. Megaritis (2015)

Colloquium Mathematicae

In this paper, on the family (Y) of all open subsets of a space Y we define the so called quasi Scott topology, denoted by τ q S c . This topology defines in a standard way, on the set C(Y,Z) of all continuous maps of the space Y to a space Z, a topology t q I s called the quasi Isbell topology. The latter topology is always larger than or equal to the Isbell topology, and smaller than or equal to the strong Isbell topology. Results and problems concerning the topology t q I s are given.

The quasi topology associated with a countably subadditive set function

Bent Fuglede (1971)

Annales de l'institut Fourier

This is a general study of an increasing, countably subadditive set function, called a capacity, and defined on the subsets of a topological space X . The principal aim is the study of the “quasi-topological” properties of subsets of X , or of numerical functions on X , with respect to such a capacity C . Analogues are obtained to various important properties of the fine topology in potential theory, notably the quasi Lindelöf principle (Doob), the existence of a fine support (Getoor), and the theorem...

The rank of the diagonal and submetrizability

Aleksander V. Arhangel'skii, Raushan Z. Buzyakova (2006)

Commentationes Mathematicae Universitatis Carolinae

Several topological properties lying between the submetrizability and the G δ -diagonal property are studied. We are mostly interested in their relationship to each other and to the submetrizability. The first example of a Tychonoff space with a regular G δ -diagonal but without a zero-set diagonal is given. The same example shows that a Tychonoff separable space with a regular G δ -diagonal need not be submetrizable. We give a necessary and sufficient condition for submetrizability of a regular separable...

The regular topology on C ( X )

Wolf Iberkleid, Ramiro Lafuente-Rodriguez, Warren Wm. McGovern (2011)

Commentationes Mathematicae Universitatis Carolinae

Hewitt [Rings of real-valued continuous functions. I., Trans. Amer. Math. Soc. 64 (1948), 45–99] defined the m -topology on C ( X ) , denoted C m ( X ) , and demonstrated that certain topological properties of X could be characterized by certain topological properties of C m ( X ) . For example, he showed that X is pseudocompact if and only if C m ( X ) is a metrizable space; in this case the m -topology is precisely the topology of uniform convergence. What is interesting with regards to the m -topology is that it is possible, with...

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