Wallman's method
Infinite lower triangular matrices of generalized Schröder numbers are used to construct a two-parameter class of invertible sequence transformations. Their inverses are given by triangular matrices of coordination numbers. The two-parameter class of Schröder transformations is merged into a one-parameter class of stretched Riordan arrays, the left-inverses of which consist of matrices of crystal ball numbers. Schröder and inverse Schröder transforms of important sequences are calculated.
We effectively construct in the Hilbert cube two sets with the following properties: (a) , (b) is discrete-dense, i.e. dense in , where denotes the unit interval equipped with the discrete topology, (c) , are open in . In fact, , , where , . , are basic open sets and , , (d) , is point symmetric about . Instead of we could have taken any -space or a digital interval, where the resolution (number of points) increases with .
We establish an identity between Delannoy numbers and tetrahedral numbers of arbitrary dimension.
Let be the large source of epimorphisms in the category of Urysohn spaces constructed in [2]. A sink is called natural, if for all . In this paper natural sinks are characterized. As a result it is shown that permits no -factorization structure for arbitrary (large) sources.
The structure of sub-, pseudo- and quasimaximal spaces is investigated. A method of constructing non-trivial quasimaximal spaces is presented.
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