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Topologies and smooth maps on initial and final objects in the category of Frolicher spaces

100%

Batubenge A. , Tshilombo H.

2009

tom Vol. 42, nr 3

641-655

In this paper, we show that when the Frolicher smooth structure is induced on a subset or a quotient set, there are three natural topologies underlying the resulting object. We study these topologies and compare them in each case. It is known that the topology generated by strucure functions is the weakest one in which all functions and curves on the space are continuous. We show that on a subspace, it is rather the trace topology which has this property, while the three topologies are coincident on the quotient space. We construct a base for the Frolicher topology and using either a base or a subbase in the sense of A. Frolicher [9], we characterise the morphisms of this category.

K-connectivity

88%

Jędrzejewski J. , Kowalczyk S.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2008

tom Vol. 13

27-32

The notion of some connectedness is introduced and it is compared with usual connected sets. Subsets of a topological space that are called i-connected have been introduced by J. Knop & M. Wróbel in [2] in 2006 The reason of introducing this notion is based upon the fact that a subset of the space R with nonempty interior is i-connected if and only if it is connected.

On cluster sets of connected functions

75%

Jędrzejewski J.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2009

tom Vol. 14

51-61

In the article we discuss some properties of a few classes of connected functions and cluster sets of functions from those classes.

On gβ-regular and gβ-normal spaces

63%

Tahiliani S.

2007

tom Vol. 40, nr 3

739-743

The aim of this paper is to introduce and study two new classes of spaces, called gβ-regular and gβ-normal spaces. The concept of gβ-regularity and gβ-normality are separation properties obtained by utilizing gβ-closed sets. Recall that a subset A of a space (X, r ) is called gβ-closed if β Cl(A) is a subset of U where A is a subset of U and U is open in X.

Applications of topology in computer algorithms

44%

Telgarsky R.

Matematyka Stosowana : matematyka dla społeczeństwa

2011

tom T. 13/54

35-46

The aim of this paper is to discuss some applications of general topology in computer algorithms including modeling and simulation, and also in computer graphics and image processing. While the progress in these areas heavily depends on advances in computing hardware, the major intellectual achievements are the algorithms. The applications of general topology in other branches of mathematics are not discussed, since they are not applications of mathematics outside of mathematics.

Celem niniejszego artykułu jest omówienie niektórych zastosowań topologii ogólnej w algorytmach komputerowych w kontekście modelowania i symulacji, a także w grafice komputerowej i przetwarzaniu obrazów. Choć postęp w tych dziedzinach jest bardzo uzależniony od postępów w sprzęcie komputerowym, jednak głównymi osiągnięciami intelektualnymi są algorytmy. Zastosowania topologii ogólnej w innych działach matematyki nie są omawiane, ponieważ nie są one zastosowaniami matematyki poza matematyką.