Advanced Topology-1 covers a variety of concepts and properties related to semi-open sets, mappings, continuity, and various types of topological spaces.
Let’s break down and explain each part:
- Semi-Open Sets and Semi-Closed Sets:
- Semi-Open Sets: Sets that are both open and closed in some sense.
- Semi-Closed Sets: Sets that are both closed and not open. Elements in a semi-closed set may or may not be included in the boundary.
- Semi-Open, Semi-Closed Mappings:
- Mappings that preserve semi-open or semi-closed sets under inverse image.
- Almost Closed and Almost Open Mappings:
- Mappings that are nearly closed or nearly open, preserving certain properties.
- s-Continuous, s-Open, and s-Closed Functions:
- s-Continuous Functions: Functions that preserve certain types of sets.
- s-Open and s-Closed Functions: Functions that behave analogously to semi-open and semi-closed sets.
- Semi-Closure and Semi-Interior:
- Semi-Closure: An operation similar to closure but not necessarily capturing all limit points.
- Semi-Interior: Similar to interior but not necessarily capturing all interior points.
- Weakly Continuous and Almost Continuous Mappings:
- Mappings that are nearly continuous or satisfy a weaker form of continuity.
- Semi-Weekly Continuous Mappings and s-Connectedness:
- Mappings and spaces that exhibit a form of continuity and connectedness that is less stringent than traditional notions.
- Strongly Continuous Mapping:
- Strongly Continuous Mapping: A more stringent form of continuity.
- Semi-T Spaces, i=0,1,2 and Their Properties:
- Semi-T Spaces: Topological spaces with certain properties related to the T separation axiom.
- i=0,1,2: Different levels of separation and their associated properties.
- Semi-Regular, Almost Regular, Almost Completely Regular, Semi-Normal, Almost Normal Spaces:
- Spaces that satisfy particular separation properties have varying degrees of regularity.
- Regular Closed Functions and Hausdorff Spaces:
- Functions that preserve certain types of sets and spaces satisfy the Hausdorff separation axiom.
- Irresolute and Almost Irresolute Functions:
- Functions that are nearly irresolute or satisfy a weaker form of irresolution.
- s-Weakly Hausdorff Spaces, s-Normal, s-Regular Spaces and Their Properties:
- Spaces that satisfy certain properties related to weak forms of separation and regularity.
Keep visiting our website www.RanaMaths.com
Advanced Topology-1