Bases and Subbases

  • Local Base for a Topology

    Let be a topological space and , then the sub collection is said to be local bases at a point , if belonging to an open set , there exist a member of , such that . It can be defined as, let be a topological space and . A sub collection of is said […]

  • Subbase for a Topology

    Let be a topological space. A sub-collection of subset of is said to be an open subbase for or a subbase for topology if all finite intersection of members of forms a base for . In others words, A class of open sets of a space is called a subbase for a topology on , […]

  • Base or Open Base of a Topology

    Let be a topological space, then the sub collection of is said to be base or bases or open base for , if each member of can be expressed as union of members of . In other words let be a topological space, then the sub collection of is said to be base, if for […]

  • First Countable Space

    Let be a topological space, then is said to be first countable space if for every has a countable local bases, i.e., if every , is countable. In other words, a topological space is said to be the first countable space if every point of has a countable neighbourhood base. A first countable space is […]

  • Second Countable Space

    Let be a topological space, then is said to be second countable space, if has a countable bases. In other words, a topological space is said to be second countable space if it has a countable open base. A second countable space is also said to be a space satisfying the second axiom of countability. […]

  • Lindelof Space

    Open Cover: Let be a topological space. A collection of open subsets of is said to be open cover for if . A sub-collection of an open cover which is itself an open cover is called a sub-cover. Lindelof Space: A topological space is said to be a Lindelof space if every open cover of […]

  • Separable Space

    A topological space , is said to be separable space, if it has a countable dense subset in . i.e., , , or , where is an open set. In other words, a space is said to be separable space if there is a subset of such that (1) is countable (2) ( is dense […]

  • Open Mapping and Closed Mapping

    Open Mapping: A mapping from one topological space into another topological space is said to be an open mapping if for every open set in , is open in . In other words a mapping is open if it carries open sets over to open sets. A continuous mapping may not be open. Example: Let […]

  • Continuity in Topological Spaces

    Let be a function define from topological space to topological space , then is said to be continuous at a point if every neighbourhood of , there exist a neighbourhood of , such that . In other words, let be two topological spaces. A function is said to be continuous at a point if and […]

  • Homeomorphism

    A function is said to be homeomorphism (topological mapping) if and only if the following conditions are satisfied: (1) is bijective. (2) is continuous. (3) is continuous. It may be noted that if is a homeomorphism from to , then is said to be homeomorphic to and is denoted by . Form the definition of […]

  • Topological Property

    A property is said to be a topological property if whenever a space has the property , all spaces which are homeomorphic to also have the property , . In other words, a topological property is a property which if possessed by a topological space is also possessed by all topological spaces homeomorphic to that […]

  • Product Topology

    Products of Sets: If and are two non-empty sets, then Cartesian product is defined as . Projection Maps: Let and be non-empty sets, then it can be defined the following two functions (1) defined as for all (2) defined as for all . The above maps are called the projection maps on and respectively. Note: […]