Topological Spaces

  • Definition of Topology

    Let be a non empty set. A collection of subsets of is said to be a topology on if the following conditions are satisfied: (i) The union of any number of members of belongs to . (ii) The intersection of finite number of members of belongs to. (iii) The empty set and set itself belongs […]

  • Coarser and Finer Topology

    If and are two topologies defined on the non empty set X such that , i.e. each member of is also in , then is said to be coarser or weaker than and is said to be finer or stronger than . It may be noted that indiscrete topology defined on the non empty set […]

  • Indiscrete and Discrete Topology

    Indiscrete Topology: The collection of the non empty set and the set X itself is always a topology on X and is called the indiscrete topology on X. In other words, for any non empty set X, the collection is an indiscrete topology on X, and the space is called the indiscrete topological space or […]

  • Intersection of Topologies

    Intersection of any two topologies on a non empty set is always topology on that set. While the union of two topologies may not be a topology on that set. Example: Let is a topology on X. is not a topology on X. Given two (and in fact any number of) topologies , on X […]

  • Usual Topology on Real

    Usual Topology on : A collection of subsets of which can be can be expressed as union of open intervals, forms a topology on , and is called topology on . Remark: Every open interval is an open set but the converse may not be true. Usual Topology on : Consider the Cartesian plane , […]

  • Open Subset of a Topological Space

    Let be a topological space, then a member of is said to be an open set in . Thus, in a topological space , the members of are said to be open subsets of . Since and full space are always the member of , so and are always open sets in . On the […]

  • Cofinite Topology

    Let is a non empty set, and then the collection of subsets of whose compliments are finite along with (empty set), forms a topology on , and is called co-finite topology. Example: Let with topology is a co – finite topology because the compliments of all the subsets of are finite. Note: It may be […]

  • Closed Subset of a Topological Space

    Let be a topological space, then a subset of X whose complement is a member of is said to be a closed set in X. Thus, in a topological space , the complements of the members of are said to be closed subsets of X. Since and the full space X are always closed sets […]

  • Subspaces of Topology

    We shall describe a method of constructing new topologies from the given ones. If is a topological space and is any subset, there is a natural way in which can “inherit” a topology from the parent set . It is easy to verify that the set , as runs through , is a topology on […]

  • Limit Point of a Set

    Let is a topological space with topology , and is a subset of . A point is said to be the limit point or accumulation point or cluster point of if each open set containing contains at least one point of different from . In other words, a point of a topological space is said […]

  • Isolated Point of a Set

    Let be a subset of a topological space , then a point is said to be an isolated point of if there exist an open set containing which does not contain any point of different from . In other words, a point is said to be an isolated point of if there exist an open […]

  • Closure of a Set

    Let be a topological space and is a subset of , then the closure of is denoted by or is the intersection of all closed sets containing or all closed super set of . i.e. the smallest closed set containing . On the other hand it can also be as let be a topological space […]