Disconnectedness, completeness and compactness.

1,413

(I'll stick to examples within the real numbers)

Complete but not compact: the reals work, and are a fine example. Or otherwise $[1,\infty)$, e.g., or that set union some compact set, etc.

Not complete, disconnected: the rationals (usual metric) are fine. Or the irrationals, in the usual metric. Or $(0,1) \cup \{2\}$ will work as well.

Connected, not complete: Indeed $(0,1)$ is an example of this.

Compact but not connected: $[0,1] \cup [2,3]$ is a simple example, the Cantor set (if you know it) is a more complex one (which is totally disconnected, i.e. has no connected subsets except the singletons),

Complete, not disconnected = Complete and connected: the reals themselves, or $[0,1]$, or $[1,\infty)$.

Share:
1,413
Kavita
Author by

Kavita

Updated on August 01, 2022

Comments

  • Kavita
    Kavita over 1 year

    I am in search of examples of metric space which is

    1) Complete but not compact

    2) Not complete but disconnected

    3) Connected but not Complete

    4) Compact but not connected.

    5) Complete but not disconnected.

    I assure the users that this question is not at all a homework, but I am interested to find such examples. For 1) I have R with usual metric and infinite metric space.

    For 2) I have set of rational numbers with usual metric.

    For 3) 4) 5), I am not sure about my answers.

    I know Q, (0,1) are incomplete metric spaces.

    I need some basic examples so that I can understand easily. Also don't have enough examples of connected metric space.

  • Fareed Abi Farraj
    Fareed Abi Farraj almost 2 years
    Is there a complete not connected metric space?
  • Henno Brandsma
    Henno Brandsma almost 2 years
    @FareedAbiFarraj yes, the discrete space is the simplest example.