• Kyungpook Mathematical Journal
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• v.27 no.1
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• pp.27-33
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• 1987

Minimal s-Urysohn and minimal s-regular spaces are studied. An s-Urysohn (respectively, s-regular) space (X, $\mathfrak{T}$) is said to be minimal s-Urysohn (respectively, minimal s-regular) if for no topology $\mathfrak{T}^{\prime}$ on X which is strictly weaker than $\mathfrak{T}$, (X, $\mathfrak{T}^{\prime}$) is s-Urysohn (respectively s-regular). Several characterizations and other related properties of these classes of spaces have been obtained. The present paper is a study of minimal P-spaces where P refers to the property of being an s-Urysohn space or an s-regular space. A P-space (X, $\mathfrak{T}$) is said to be minimal P if for no topology $\mathfrak{T}^{\prime}$ on X such that $\mathfrak{T}^{\prime}$ is strictly weaker than $\mathfrak{T}$, (X, $\mathfrak{T}^{\prime}$) has the property P. A space X is said to be s-Urysohn [2] if for any two distinct points x and y of X there exist semi-open set U and V containing x and y respectively such that $clU{\bigcap}clV={\phi}$, where clU denotes the closure of U. A space X is said to be s-regular [6] if for any point x and a closed set F not containing x there exist disjoint semi-open sets U and V such that $x{\in}U$ and $F{\subseteq}V$. Throughout the paper the spaces are assumed to be Hausdorff.

• Communications of the Korean Mathematical Society
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• v.29 no.4
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• pp.555-568
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• 2014

A set with an operation defined on a family of subsets is studied. The operation is used to generalize the topological space itself. The operation defines the operation-open subsets in the set. Relations are studied among two types of the interiors and the closures of subsets. Some properties of maximal operation-open sets are obtained. Semi-open sets and pre-open sets are defined in the sets with operations and some relations among them are proved.

• The Pure and Applied Mathematics
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• v.29 no.1
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• pp.93-102
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• 2022

The impact of programming languages in the research sector has helped lot of researchers to broaden their view and extend their work without any limitation. More importantly, even the complex problems can be solved in no matter of time while converting them into a programming language. This convenience provides upper hand for the researchers as it places them in a comfort zone where they can work without much stress. With this context, we have converted the research problems in Topology into programming language with the help of Python. In this paper, we have developed a Python program to find the weaker form of closed sets namely alpha closed set, semi closed set, pre closed set, beta closed set and regular closed set.

• Journal of the Korean Institute of Landscape Architecture
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• v.32 no.3
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• pp.1-17
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• 2004

The focus of this study is the promotion of green area volumes and their naturalness, water circulation system, decline of entropy, creation of biological habitats and linkage of separated urban green space. Re-presentative urban biotope survey sites were categorized as urban biotope, semi-natural biotope, and natural forest. In the urban biotope, a residential biotope was constructed near the Han river and in mountain areas. The green-area ratio at the housing complex was about 25%. GVZ(Grunvolumenzahl) was 0.35m$^3$/m$^2$ at the 5∼10-story housing complex, and 1.53m$^3$/m$^2$ over the 11-story. As for the green-area structure of the housing complex, canopy layer, understory layer, and shrub layer were not differentiated and the green-area volume was not high enough. The green-area ratio of school areas as a public area biotope was 5∼20%. GVZ was 1.12m$^3$/m$^2$ at Myungduk High School, and 1.78m$^3$/m$^2$ at Jeonggok Elementary School. In order to convert the urban biotope into an ecological area, green areas around the buildings should be connected to urban buffer green areas, and multi-layer structures should be established with natural plant species. In the semi-natural biotope, neighbor parks were created park in the vicinity of the natural forests. GVZ was 0.28m$^3$/m$^2$, and plantation was established with single layer structure and was definitely insufficient for the area. The urban buffer green areas have been established in strip corridors with the width of 20∼123m. In those areas, GVZ was 0.16∼0.27m$^3$/m$^2$ and had a deficient canopy layer, understory layer, and shrub layer. Soil conditions were not favorable for tree growth. In the natural biotope, GVZ of the plantation was 1.03∼1.5m$^3$/m$^2$ but the high crown closure of this area reduces the chance of species change and succession. GVZ of natural forest was 2.53∼2.57m$^3$/m$^2$. It is desirable to plant diverse plants and the natural forest should be succeeded by broad-leaf deciduous tree species. To improve the value of biotope at Kangseo-Gu, building height needs to be limited to reduce the environmental deterioration in the city. In order to maintain the water circulation system, water-permeable material is recommened when the urban surface areas are paved. The establishment of a water circulation system will improve ground water levels, soil moisture, water quality, and habitats. In order to improve biological diversity, it is desirable to have multi-layer structures in urban green areas with native species.