Degrees of <i>L</i>-Continuity for Mappings between <i>L</i>-Topological Spaces

• Main Authors: Zhenyu Xiu, Qinghua Li
• Format: Article
• Language: English
• Published: MDPI AG 2019-10-01
• Series: Mathematics
• Subjects: <i>l</i>-cotopological space; <i>l</i>-topological space; degree of <i>l</i>-continuity; degree of <i>l</i>-closedness; degree of <i>l</i>-openness
• Online Access: https://www.mdpi.com/2227-7390/7/11/1013

By means of the residual implication on a frame <i>L</i>, a degree approach to <i>L</i>-continuity and <i>L</i>-closedness for mappings between <i>L</i>-cotopological spaces are defined and their properties are investigated systematically. In addition, in the situation of <i>L</i>-topological spaces, degrees of <i>L</i>-continuity and of <i>L</i>-openness for mappings are proposed and their connections are studied. Moreover, if <i>L</i> is a frame with an order-reversing involution <inline-formula> <math display="inline"> <semantics> <msup> <mrow></mrow> <mo>&#8242;</mo> </msup> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>b</mi> <mo>&#8242;</mo> </msup> <mo>=</mo> <mi>b</mi> <mo>&#8594;</mo> <mo>&#10178;</mo> </mrow> </semantics> </math> </inline-formula> for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mo>&#8712;</mo> <mi>L</mi> </mrow> </semantics> </math> </inline-formula>, then degrees of <i>L</i>-continuity for mappings between <i>L</i>-cotopological spaces and degrees of <i>L</i>-continuity for mappings between <i>L</i>-topological spaces are equivalent.