Ordinary Topological Spaces Research Articles

Some more results on induced fuzzy topological spaces

The concept of induced fuzzy topological space, introduced by Weiss [J. Math. Anal. Appl. 50 (1975) 142–150], was defined with the notions of a lower semi-continuous function. The purpose of this paper is to present some new results on induced fuzzy topological spaces with the notion of α-open sets introduced by Njastad [Pacific J. Math. 15 (1965) 961–970]. We also study the connections between some covering properties of an ordinary topological space ( X, T) and its corresponding induced fuzzy topological space ( X, W( T)). In the last section we also introduce a new class of fuzzy supra topological spaces - α-induced fuzzy supra topological space (α-IFST space) and study the properties.

Jordan surfaces in simply connected digital spaces

Certain tasks in multidimensional digital image analysis, in particular surface detection and volume estimation, lead us to the study of “surfaces” in the digital environment. It is desirable that these surfaces should have a “Jordan” property analogous to that of simple closed curves in two dimensions-namely, that they should partition the underlying space into an inside and an outside which are disconnected from each other. A version of this property, called near-Jordanness, has been previously defined and has been shown to be useful within a general theory of surfaces in digital spaces. The definition of a near-Jordan surface is global: it demands that all paths from the interior to the exterior cross the surface. This makes it difficult in many practical applications to check whether surfaces are near-Jordan. The work reported in this paper is motivated by the desire for a “local” condition, such that if a surface satisfies this condition at each of its elements, then it is guaranteed to be near-Jordan. In the search for such a condition, we were led to a concept of “simple connectedness” of a digital space, which resembles simple connectedness in ordinary topological spaces. We were then able to formulate the desired local condition for simply connected digital spaces. Many digital spaces, in particular those based on the commonly-studied tessellations of n-dimensional Euclidean space, are shown to be simply connected and thus our theory yields general sufficient conditions for boundaries in binary pictures to be near-Jordan.

Stable domains are generalized topological spaces

We present a generalization of the notion of topological structure on a set such that DI- or (L-) domains can be naturally given such topologies in a way making the continuous functions exactly the stable functions. As ordinary topological spaces give rise to locales, these generalized spaces lead to a generalized form of locale, that we define and explore.

On L-fuzzy topological spaces

L-fuzzy sets and, subsequently, L-fuzzy topological spaces are considered in this paper, where L is a complete and completely distributive non-atomic Boolean algebra. A few separation properties as well as subspace L-fuzzy topology are defined here, in the light of an earlier paper. These are more similar to ordinary topological spaces than to fuzzy topological spaces.