Connectedness and Compactness

  • Disconnected Space

    A topological space is said to be disconnected space if can be separated as the union of two non-empty disjoint open sets. In other words, a topological space is said to be a disconnected space if there exist non-empty open sets and such that and . The pair is called the disconnection of . Example: […]

  • Connected Space

    A topological space which cannot be written as the union of two non-empty disjoint open sets is said to be a connected space. n other words, a space is connected if it is not the union of two non-empty disjoint open sets. Example: very indiscrete space is connected. Let be an indiscrete space, then is […]

  • Components of a Space

    A connected subspace of a topological space is said to be the component of if it is not properly contained in any connected subspace of . Note: The component of a connected space is the whole space, , itself. Example: The singleton subset of a two-point discrete space is its components. Let is be two-point […]

  • Totally Disconnected Space

    A topological space is said to be totally disconnected space if any pair of distinct of can be separated by a disconnection of . In other words, a topological space is said to be totally disconnected space if for any two points and of , there is a disconnection of such that and . In […]

  • Some Applications of Connectedness

    In some applications of connectedness, we shall define two fixed point theorems in connection with application of connectedness. Fixed point theorems are useful in obtaining the (unique) solutions of differential and integral equations. Fixed Point: Let be a self mapping. A point is called a fixed point of , if . It may be noted […]

  • Compact Space

    Cover and Sub-Cover: Let be a topological space. A collection of subsets of is said to be a cover of if . A sub-collection of is said to be a sub-cover of if itself is a cover of . Open Cover and Open Sub-Cover: Let be a topological space. A collection of open subsets of […]

  • Finite Intersection Property

    A collection of subsets of a non-empty set is said to have the finite intersection property if every finite sub-collection of has non-empty intersection. In other words, the collection of subsets of the topological space is said to have finite intersection property if every finite sub-collection of has non-empty intersection, i.e. for any finite subset […]

  • Local Compact

    A space is said to be locally compact (briefly Compact) at if and only if has a compact neighbourhood in . If is compact at every point, then is called a locally compact space. Examples: • Compact spaces are compact. Suppose is compact, is a neighbourhood of each of its points implies is compact. • […]