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In this study, we generalized two known open sets called semi-open and h-open sets to find a new type called semi-h-open set in topological space We have presented the relationship of several famous open sets to this set. We have also studied the semi-h-open continuity in topological space. In this study, we generalized two known open sets called semi-open and h-open sets to find a new type called semi-h-open set in topological space We have presented the relationship of several famous open sets to this set. We have also studied the semi-h-open continuity in topological space. we generalized two known open sets.

In this paper, we propose a unified concept encompassing generalizations of two types of families defined based on Levine's notions of generalized closed sets and Maki's  Λ  sets. The methods used in this investigation are described in my previous work, where a unified concept of general closedness is presented. From a methodology point of view, the present concept is symmetric to the previous. In generalizing open subsets, one can use the two methods. According to the first one, the family of Levine's generalization is used as some base to build the family of closed subsets of the new topology. In the second method, the family of open subsets is extended, in the same way, as the family of closed subsets in the classic Levine's method. The results obtained in this general conception easily extend and imply well-known theorems of this area of investigation. In the literature on this issue, many versions of generalizations of  Λ -sets have been investigated. The tools used in this paper enabled us to prove that there exist at most 10 generalizations of these types, and we show the relationships between them in the graph. As a result, it turns out that some generalizations investigated in the literature are trivial. [ABSTRACT FROM AUTHOR]

The paper initially investigates semi-topological properties of the Khalimsky (K-, for brevity) topological version of the classical Jordan curve theorem. Assume a simple closed K-curve with l elements in (Z 2 , κ 2) that is the K-topological plane, denoted by C K l for brevity. Then every C K l separates ( Z 2 , κ 2) into exactly two nonempty components that may be neither open nor closed in ( Z 2 , κ 2) . Hence we need to investigate semi-topological features of C K l and Z 2 \ C K l in (Z 2 , κ 2) . We first show that not every C K l is always semi-open or semi-closed in (Z 2 , κ 2) . Second, we find a condition for C K l to be either semi-open or semi-closed in (Z 2 , κ 2) . After establishing a continuous analog of C K l denoted by A (C K l) (⊂ R 2) , we show that A (C K l) is both semi-open and semi-closed in (R 2 , U) that is the 2-dimensional real plane with the usual topology. Besides, we show that A (C K l) always separates (R 2 , U) into two non-empty components, denoted by C and D, that are both semi-open and semi-closed to obtain a partition of R 2 , i.e., { C , D , A (C K l) } . Finally, we obtain a partition of Z 2 , i.e., { I (C K l) , O (C K l) , C K l } and prove that each of I (C K l) and O (C K l) is semi-closed and it need not be semi-open, where I (C K l) and O (C K l) are called an inside and outside of C K l , respectively. [ABSTRACT FROM AUTHOR]

• The prototype varies within 15 % in ambient temperature range of −30 ∼ 7℃. • The frosting has no impact on the overall efficiency of system. • It has the potential for application in the field of high temperature heating. • Utilizing high-efficiency compressors could improve the system's coefficient of performance by 13.47 ∼ 31.50 %. Different from conventional vapor compression heat pump system, this investigation presented a semi-open air cycle heat pump system, whose performance was experimentally assessed across diverse conditions. The prototype showcased stable heating performance with coefficient of performance surpassing 1.0 at ambient temperatures above −7°C and maintaining its variation within 15 % in the ambient temperature range of (−30 ∼ 7) °C. According to the analysis of the experimental results, although frosting has an impact on some parameters, the overall efficiency of the system remains stable over time. These special designs of high-pressure ratio and oil cooling of compressor could make the system temperature reaching 132 °C, indicating that the system has the potential for application in the field of high temperature heating. Heat exchanger analysis revealed an imbalance in heat exchange ratios between oil–water heat exchanger and air–water heat exchanger. The prototype achieved 58 ∼ 72 % of a significant conversion efficiency of electric energy to oil internal energy, proposing that utilizing high compression efficiency compressors could enhance system coefficient of performance by (13.47 ∼ 31.50)%. The isentropic efficiency and expansion work recovery efficiency of booster turbo-expander were calculated as (59 ∼ 65)% and (33 ∼ 40)% respectively. Additionally, it was found that frosting could optimize the recovery rate of expansion work under specific conditions. [ABSTRACT FROM AUTHOR]